Explanatory Nonmonotonic Reasoning

Explanatory Nonmonotonic Reasoning Advances in Logic Series Editor: Dov M Gabbay FRSC FAvH Department of Computer Sci...

Author: Alexander Bochman

106 downloads 911 Views 19MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Explanatory Nonmonotonic Reasoning

Advances in Logic Series Editor: Dov M Gabbay FRSC FAvH Department of Computer Science King's College London Strand, London WC2R 2LS UK [email protected] Published Vol. 1

Essays on Non-Classical Logic by H. Wansing

Vol. 2

Fork Algebras in Algebra, Logic and Computer Science by M. Frias

Vol. 3

Reasoning about Theoretical Entities by T. Forster

Vol. 4

Explanatory Nonmonotonic Reasoning by A. Bochman

Advances in Logic - Vol. 4


Alexander Bochman Holon Academic Institute of Technology, Israel

Y ^ World Scientific NEW JERSEY • LONDON

• SINGAPORE

• BEIJING • S H A N G H A I • H O N G K O N G

• TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

EXPLANATORY NONMONOTONIC REASONING Advances in Logic — Vol. 4 Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

Forphotocopying of material in this volume, please pay acopying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-256-101-3

Printed in Singapore by World Scientific Printers (S) Pte Ltd

To Jonathan Stavi and my family.

Preface

Like my preceding book, [Bochman, 2001], the present study is also a systematic attempt to answer the question 'what is nonmonotonic reasoning?'. It complements the previous book by giving a logical formalization to the original approach to nonmonotonic reasoning that includes default logic, autoepistemic and modal nonmonotonic logics, and logic programming. We call this approach explanatory nonmonotonic reasoning, since the notion of explanation can be seen as the ultimate and unifying basis behind these nonmonotonic formalisms. Three aspects distinguish this book from previous studies in this area. First, the book provides a uniform generalized theory of explanatory nonmonotonic reasoning rather than a description of existing nonmonotonic logics. Though the latter are shown to be covered by this theory, the formalism of biconsequence relations, taken as a logical basis of this study, suggests a powerful generalization going in most cases far beyond existing nonmonotonic formalisms. Second, the book shifts attention to some relatively recent, non-epistemic approaches to nonmonotonic reasoning, such as four-valued biconsequence relations, causal reasoning and argumentation theory. These formalisms actually fill the gap between logic programming, on the one hand, and traditional nonmonotonic logics, on the other. Accordingly, default and modal nonmonotonic logics are covered only in the last two chapters, and only as parts of the more general formalism of epistemic biconsequence relations. Last but not least, the book focuses on the logical, monotonic basis of explanatory nonmonotonic reasoning. In this sense, it is as much about logic as it is about nonmonotonic reasoning. As the main benefit of this approach, it will be shown that different formalisms of explanatory nonmonotonic reasoning are based on essentially the same principles and models, the main distinction being the underlying logical vii

viii


formalisms that host such a reasoning. The intended audience of this book consists of graduate students and researchers in AI, on the one hand, and general logicians, on the other. In fact, one of the aims of the book consists in persuading the former about the need of logic, and the latter that nonmonotonic reasoning should be an integral part of general logical research. A. Bochman

Contents

vii

Preface 1. Introduction

2.

1

1.1 Two Theories of Nonmonotonic Reasoning 1.2 Explanatory Nonmonotonic Reasoning 1.3 Explanatory Nonmonotonic Reasoning and Logic 1.3.1 Logic in logic programming 1.3.2 A logical basis of nonmonotonic logics 1.3.3 Objective vs. epistemic formalisms 1.4 Overview of the Book

1 4 6 8 9 12 14

Scott Consequence Relations

19

2.1 Basic 2.1.1 2.1.2 2.1.3 2.1.4

19 21 22 22 23 24 26 28 29 32 34 39 40 41

Definitions Tarski consequence relations Representation theorem Compact sets of theories Sequent theories 2.1.4.1 Dependencies 2.1.5 Relativized consequence relations 2.1.6 The classical sequent calculus 2.2 Minimal and Supported Theories 2.2.1 Completion and loop-completion 2.3 Unitary Consequence Relations and Their Theories 2.4 Circumscription 2.4.1 Partial deduction 2.4.2 Compact circumscription ix

x

3.

4.


2.5 Supraclassicality 2.5.1 Minimal and classically supported theories

44 47

Biconsequence Relations

51

3.1 Biconsequence Relations 3.1.1 Semantics 3.1.2 Duality 3.1.3 Bisequent theories 3.1.4 Four-valued representation 3.2 Structural Rules 3.2.1 Consistency 3.2.2 Completeness 3.2.3 Scott biconsequence relations 3.2.3.1 Scott-equivalence 3.2.4 Ordering 3.2.5 Regularity 3.2.6 Semi-classicality and tightness 3.2.7 Invariance 3.3 Restricted Kinds of Biconsequence Relations 3.3.1 Default biconsequence relations 3.3.1.1 Positive biconsequence relations 3.3.2 Autoepistemic biconsequence relations 3.3.2.1 Minimal equivalence 3.3.2.2 Autoepistemic reduction and GPPE 3.3.3 Simple biconsequence relations 3.4 Circumscribed Biconsequence Relations

52 54 56 56 58 60 60 62 63 64 65 66 69 71 73 75 77 78 79 80 81 82

Four-Valued Logics

85

4.1 Introducing Connectives 4.1.1 Classical four-valued connectives 4.1.2 Local classical connectives 4.1.3 Invariant connectives 4.1.4 Conservative connectives 4.1.5 Normal forms 4.1.6 Alternative representations 4.1.7 Classical translation 4.2 Four-valued Entailment and 4-Theories 4.3 Invariant Logic

85 88 93 94 95 97 99 100 102 105

Contents

5.

6.

xi

4.3.1 Axiomatic representation 4.3.2 Possible worlds semantics 4.3.3 Invariant inference 4.4 Conservative and Three-Valued Logics 4.5 Coherence

105 107 110 112 114

Nonmonotonic Semantics

121

5.1 Exact Semantics 5.1.1 Exact completions 5.1.2 Minimality 5.2 Singular Semantics 5.3 Default Semantics 5.3.1 Stable completion 5.3.2 Minimal extensions 5.4 Supported Semantics 5.4.1 Completion 5.5 Partial Nonmonotonic Semantics 5.5.1 Doubling 5.5.2 Partial exact semantics 5.5.3 Partial default semantics 5.5.4 Partial supported semantics 5.6 Bisequent Interpretations of General Logic Programs . . . . 5.6.1 A default interpretation 5.6.2 An autoepistemic interpretation 5.7 Nonmonotonic Completion and Weak Semantics 5.7.1 Nonmonotonic completion 5.7.2 Weak semantics 5.7.2.1 Well-founded and static semantics 5.7.2.2 Local completions: stationary semantics and stable classes 5.7.2.3 Invariant completion and partial stable semantics 5.7.2.4 Classical completion and stable semantics .

121 126 128 129 132 136 136 139 143 144 147 149 152 154 155 156 158 163 163 165 166

Default Consequence Relations

171

6.1 Default Consequence Relations 6.1.1 Semantics 6.1.2 Default theories

172 174 175

167 169 169

xii


6.2 Extensions and Expansions 6.3 Partial Expansions and Extensions 6.3.1 Doubling 6.4 Kinds of Default Reasoning 6.5 Default Scott Consequence Relations 6.6 Reflexive Default Consequence Relations 6.6.1 Regular default consequence relations 6.7 Autoepistemic Consequence Relations 6.8 Default Consequence Relations and Normal Programs . . . 7. Argumentation Theory 7.1 Collective Argumentation 7.1.1 Argument theories 7.1.2 Four-valued semantics 7.2 Argumentation vs. (Bi)consequence relations 7.3 Main Kinds of Argumentation 7.3.1 Classical argumentation 7.3.2 Negative argumentation 7.3.3 Positive argumentation 7.3.3.1 Normal positive argumentation 7.3.3.2 Consistent attack relations 7.3.3.3 Weak positivity 7.4 Nonmonotonic Semantics of Argumentation 7.4.1 Normal (Dung) argumentation 7.4.2 Stable and partial stable semantics 7.4.3 Admissibility semantics 7.4.4 Argumentation in logic programming 7.5 Propositional Argumentation 8. Production and Causal Inference 8.1 Production Inference 8.1.1 Semantics 8.1.2 Causal theories 8.1.3 Regular production inference 8.1.4 Production inference vs. consequence relations 8.2 Nonmonotonic Production Semantics 8.3 Production Inference and Abduction 8.3.1 Abductive systems and abductive semantics

178 180 182 184 186 188 190 193 198 201 202 204 204 206 208 209 211 213 214 215 216 217 217 218 224 226 227 231

232 234 236 236 . . . 240 242 247 248

Contents

8.4

8.5

8.6 8.7 8.8

8.9

xiii

8.3.2 Abductive production inference 251 8.3.3 Quasi-abductive production inference 254 8.3.4 Abduction in literal causal theories 258 Basic and Causal Production Inference 260 8.4.1 Possible worlds semantics 261 8.4.1.1 D-validity 262 8.4.2 Production inference vs. biconsequence relations . . 263 8.4.2.1 Production inference as argumentation . . . 266 8.4.3 Causal inference 267 8.4.3.1 Abductive causal inference 271 Causal Nonmonotonic Semantics 273 8.5.1 Factual and explanatory content of causal theories . 276 8.5.2 Determinate theories and completions 277 8.5.3 Partial exact semantics 282 Negative Closure and Tightness 284 Quasi-Classicality 288 Partial Production Inference 292 8.8.1 Possible worlds semantics 294 8.8.2 Partial nonmonotonic semantics 295 8.8.3 Invariance 296 8.8.4 Doubling 298 8.8.4.1 Partial extensions and expansions 300 Causal Interpretation of Logic Programs 302 8.9.1 The stable interpretation 302 8.9.2 The supported interpretation 304 8.9.3 Partial causal interpretations 305 8.9.3.1 Partial supported interpretation 305 8.9.3.2 The partial stable interpretation 306

9. Epistemic Consequence Relations

309

9.1 General Epistemic Biconsequence Relations 310 9.2 Supraclassical Biconsequence Relations 312 9.2.1 Supraclassical relations and production inference . . 313 9.2.2 Saturated biconsequence relations 315 9.3 Classical Nonmonotonic Semantics 318 9.3.1 Classical expansions 321 9.3.2 Extended programs and classical negation 323 9.3.3 Complete extensions and expansions 325 9.4 Supraclassical Default Consequence Relations 328

xiv


9.4.1 Reiter's default logic 9.4.2 Kinds of supraclassical default consequence relations 9.4.2.1 Introspective consequence relations 9.4.2.2 Regular consequence relations 9.4.2.3 Saturated consequence relations 9.4.2.4 Classical autoepistemic consequence relations 9.4.3 Default relations vs. production inference 9.4.3.1 Autoepistemic representation of general productions 9.4.3.2 Default representation of causal productions 10.

Modal Nonmonotonic Logics 10.1 Modal Biconsequence Relations 10.1.1 Normal biconsequence relations 10.1.1.1 Weak, positive and reflexive introspection 10.1.2 F-biconsequence relations and S4F 10.1.3 Modal saturated biconsequence relations 10.2 Modal Nonmonotonic Semantics 10.2.1 Modal extensions 10.2.2 Modal expansions 10.3 Modal Default Consequence Relations 10.3.1 Prime modal consequence relations 10.3.2 Nonmonotonic semantics 10.3.2.1 A^-extensions 10.3.2.2 Modal expansions

331 332 332 334 335 336 338 338 339 341

343 346 . 350 357 361 366 368 373 377 382 387 387 390

Bibliography

395

Index

403

Chapter 1

Introduction

In this introductory chapter I will delineate the subject of this study, and the questions it is going to answer. 1.1

Two Theories of Nonmonotonic Reasoning

Studies in nonmonotonic reasoning have given rise to two basically different approaches that will be called, respectively, preferential and explanatory nonmonotonic reasoning, with little interaction between them1. The first approach encompasses nonmonotonic inference relations of [Kraus et al., 1990], and a general theory of belief change [Alchourron et al., 1985]. A detailed description of this approach can be found in [Bochman, 2001]. The second, explanatory approach is older, and it includes default and modal nonmonotonic logics, as well as logic programming with negation as failure. In fact, all the papers in the famous 1980 issue of the Artificial Intelligence Journal on nonmonotonic reasoning could be seen as belonging to this latter camp (though McCarthy's circumscription is covered also by the preferential approach). The aim of this study consists of providing a systematic logical theory for this explanatory approach to nonmonotonic reasoning. It might well be (and I certainly hope so) that both these approaches will some day become parts of a single future theory of nonmonotonic reasoning. Nevertheless, I also believe that a proper basis for a future unification should consist in a careful separation between these approaches. Such a separation, however, is not a trivial matter. The difference between the two approaches can be found on a number lr

They have been called, respectively, classical and argumentative nonmonotonic reasoning in [Bochman, 2001]. 1

2


of levels. To begin with, there are two different senses in which a logical formalism, or a reasoning system, may be called nonmonotonic. First, it may be nonmonotonic in that its rules do not admit addition of new premises, that is, the system does not allow Strengthening the Antecedent. Second, it may be nonmonotonic in the sense that adding further rules to the system may possibly invalidate some of the conclusions obtained earlier. Now, it turns out that these two kinds of nonmonotonicity are largely independent. Thus, preferential inference relations (see [Kraus et at, 1990]) are nonmonotonic in the first sense, since strengthening the antecedent does not hold for them [Birdsflydoes not imply Penguins fly). However, they are monotonic in the second sense, since addition of new preferential conditionals does not invalidate previous conclusions. On the other hand, default logic (see [Reiter, 1980]) exemplifies monotonicity of the first kind and nonmonotonicity of the second kind. Indeed, we will see that any default theory can be safely extended with default rules obtained from existing ones by strengthening their pre-requisites and justifications; such additional rules will not change the set of extensions. On the other hand, adding arbitrary new rules to the default theory may result in creating new extensions, so nonmonotonic conclusions made earlier will not, in general, be preserved. Taken by itself, however, the above distinction is a purely formal difference between formalisms, and it still does not necessarily imply that the two approaches are essentially different. Actually, one of the main incentives behind the preferential approach, already explicitly expressed in [Shoham, 1988], was the hope that default logic and other explanatory nonmonotonic formalisms can be subsumed by some generalized version of the preferential approach in the sense that extensions could be viewed as preferred models under some generalized notion of preference. Unfortunately, subsequent studies have raised grave doubts about this hope. Thus, the nonmonotonic semantics of default logic has turned out to violate even the most basic postulates of cumulative inference (see [Makinson, 1989]). A similar situation has been found in logic programming (see [Dix, 1991]). In a hindsight, this outcome should have been expected, since the selection of intended models in an explanatory approach is not preferential in a usual sense. Namely, the explanatory approach determines the intended models as models satisfying certain closure conditions with respect to the rules (see below). On a most abstract level, such models are usually expressible as fixed points of some operator which is not even monotonic. Accordingly, the supposed preference that singles out these models appears

Introduction

3

to be a trivial, zero-one preference that basically differentiates only right models from bad ones. In this sense, the above formal difference can be viewed only as a symptom of deeper conceptual distinctions between the two approaches. Both preferential and explanatory nonmonotonic reasoning can be seen as theories of a reasoned use of assumptions. Now, preferential reasoning treats such assumptions as defaults, namely as normality assumptions we can use whenever there is no evidence to the contrary. This understanding is combined with a general principle that a problem situation should be assumed as normal as it is consistently possible, given the known facts. This naturally creates a preferential setting, in which the normality of models is measured by the set of defaults they support (see [Bochman, 2001] where such a preferential order is formally described). It turns out, however, that the explanatory nonmonotonic reasoning implicitly assigns a different role to assumptions. Using the name adopted by David Poole (see [Poole, 1989; Poole, 1990]) it makes such assumptions conjectures. Conjectures are assumptions that we make in order to explain observations. The supposition of normality (or abnormality, for that matter) is not essential for such conjectures. A certain combination of symptoms may lead to a conjecture about a rare and unusual disease, while in other cases some 'ordinary' illness will suffice for explaining the observations; it seems beside the point in this case to order diseases with respect to their 'normality'. As was rightly noted by Poole, we make conjectures only if there is evidence that requires them for explanation, in contrast to defaults that can be freely assumed, unless they contradict the facts and other assumed defaults. It was also strongly argued by Poole that the distinction between normality defaults and conjectures is closely related to the distinction between prediction and explanation: while we use defaults in order to predict facts that are yet unknown, conjectures are invoked when we have to explain known facts. Unfortunately, the above presumably clear distinction has been obscured in the short history of nonmonotonic reasoning. The reason was that, from the very beginning, the main formalisms of nonmonotonic reasoning, including default logic, modal nonmonotonic logics and circumscription, have claimed their rights and responsibility on representation of normality defaults. Thus, Ray Reiter has suggested in [Reiter, 1980] that we can identify such normality assertions with a special case of default rules

4


of the form A:B/B, appropriately called normal default rules2. It is the author's conviction that the preferential approach provides a more adequate analysis of the notion of normality and, in particular, of normal defaults. All the examples in the literature that reveal a discrepancy between Reiter's normal defaults A:B/B and corresponding preferential conditionals A\^B point out in favor of the latter. In this respect, the criticism raised against default logic in the preferential camp (see, e.g., [Kraus et al., 1990; Lehmann, 1995]) seems perfectly justified, so far as we are talking only about which notion better reflects our understanding of normality. Still, there is nothing wrong or nonintuitive about having both A : B/B and A : -*B/-iB in a default theory, though it is certainly counterintuitive to treat such rules simultaneously as normality defaults. In the setting of default logic, such rules say simply that, when A holds, both B and —>B are equally admissible assumptions, or conjectures. This indicates, however, that default logic has a subject of its own that should not be extrapolated to the entire field of nonmonotonic reasoning. The above picture is of course incomplete, mainly because it stresses the differences and ignores numerous similarities and connections between the two approaches. Note, for example, that McCarthy's circumscription (with abnormality predicates) can be easily understood in both ways, which might explain why it constitutes a borderline case covered by both approaches. Speaking more generally, many reasoning tasks actually involve both these forms of nonmonotonic reasoning. Still, I believe that a clear understanding of explanatory nonmonotonic reasoning can be achieved only if it is considered as an independent reasoning paradigm with a subject, goals and principles of its own. 1.2


As the name indicates, explanation can be seen as a basic notion of an explanatory approach. Propositions and facts may be not only true or false in a model of a problem situation, but some of them are explainable by other facts and rules that are accepted. In the objective setting, some of the facts are caused by other facts and causal rules acting in the domain. Furthermore, explanatory nonmonotonic reasoning is based on very strong principles of Explanation Closure ([Schubert, 1994]) and Causal Complete2 Poole himself implicitly considered such rules as primary examples of normality defaults.

Introduction

5

ness ([Reiter, 2001]), according to which any fact holding in a model should be explained, or caused, by the rules that describe the domain. Incidentally, it is these principles that make explanatory reasoning nonmonotonic. By the above description, abductive reasoning, that is, reasoning from facts to their possible explanations, can be seen as an integral part of explanatory nonmonotonic reasoning. Ultimate explanations, or abducibles, correspond in this sense not to normality defaults, but to conjectures representing base causes that are viewed as facts that do not require explanation; we assume the latter only if we have evidence that requires them for explanation. In some domains, explanatory formalisms adopt additional, simplifying assumptions that exempt, in effect, certain classes of propositions from the burden of explanation. Reiter's Closed World Assumption (see [Reiter, 1978]) can be seen as one of the most important assumptions of this kind. According to the latter, negated propositional atoms do not require explanation. Nonmonotonic reasoning in databases and logic programming are domains for which such an assumption turns out to be most appropriate. It is important to note in this respect that the well-known minimization principle is actually a result of combining Explanation Closure with the Closed World Assumption. Indeed, we will see in what follows that semantics of logic programming, that are commonly viewed as based on the minimization principle, can be obtained directly from the combination of these two assumptions. An interesting aspect of this claim is that the minimization principle need not be viewed as a principle of scaled preference of negative information; rather, it is a by-product of the plain assertion that negated atomic propositions can be accepted without any further explanation, while positive assertions always require an explanation. As we will show in this study, the above principles will be sufficient for establishing a formal logical framework for explanatory nonmonotonic reasoning. The above description presupposes, however, a drastic departure from traditional logical representations, since it draws a richer picture of what is in the world than what is usually captured in logical models of the latter. The traditional understanding of possible worlds in logic can be traced back to Wittgenstein's Tractatus [Wittgenstein, 1961]. The atomic facts of the Tractatus' metaphysics are independent: l[e]ach item can be the case or not the case while everything else remains the same\ Consequently, there is no possible way of making an inference from one fact to another, entirely

6


different fact, and there is no causal nexus to justify such an inference. The only restriction on the structure of the world is based on the principle of non-contradiction. As a result, Wittgenstein's worlds leave no place in it for dependencies among facts and related notions, in particular for causation. Accordingly, Wittgenstein concluded that belief in such dependencies is a 'superstition'. This book is not a philosophical study, so I will refrain from indulging the reader with philosophical considerations. Still, it should be kept in mind that this book presupposes a different picture of the world, according to which the world is not a mere assemblage of unrelated facts, but something which has a structure. This structure determines dependencies among occurrent facts, while the latter serve as a basis for our explanatory and causal claims. It is this structure that makes the world intelligible and, what is especially important for this study, controllable. The picture implies, in particular, that explanatory and causal relations form an integral part of understanding of and acting in the world. Consequently, such relations should form an essential part of knowledge representation, at least in Artificial Intelligence.

1.3

Explanatory Nonmonotonic Reasoning and Logic

One of the main presuppositions of this book is that nonmonotonic reasoning should give us a more direct and adequate description of the actual ways we think about the world. In this respect, preferential nonmonotonic reasoning has had a definite advantage over traditional nonmonotonic logics in that it provided a direct semantic representation for its inference rules, namely default conditionals 'If A, then normally B\ This semantic representation allows us to assess our default claims and determines, ultimately, the actual choice of default assumptions made in particular circumstances. For a number of reasons, discussed below, the basic syntactic objects of default logic and its relatives have lacked such a clear and direct semantic interpretation. As a result, a representation of nonmonotonic reasoning problems in such systems has been largely a syntactic enterprize. Thus, in default logic the user is required to provide explicit information about when one default rule can 'block' another default (by specifying appropriate justifications in premises of such rules). This information is used as a sole factor in determining acceptable combinations of defaults. This strategy can be remarkably successful in resolving difficult cases of default interac-

Introduction

7

tion, which can be seen as the main reason why the explanatory nonmonotonic reasoning so far has had a greater impact on practical applications of nonmonotonic reasoning in AI than its preferential counterpart. Still, such a methodology puts sometimes a heavy burden on the user by requiring from him to foresee and control the results of default interaction. In other words, it does not give us a transparent and systematic way of representing empirical data, and this makes the task of knowledge representation in such systems really an art. As I see it, the main reason of these shortcomings amounts to the absence of an adequate logical basis in such nonmonotonic systems. The majority of nonmonotonic formalisms have two components. The first is an ordinary, monotonic logical framework, e.g., classical logic in circumscription, or some modal logic in modal nonmonotonic logics. The second, nonmonotonic, component involves a stipulation about which potential models should be considered as intended ones. In other words, the nonmonotonic component is built in these systems on top of a monotonic inference system. Unfortunately, such important nonmonotonic formalisms as default and autoepistemic logics, and semantics of logic programming, are usually described in a shortcut way, namely as pairs consisting of syntax and nonmonotonic semantics. To put it shortly, they are built in accordance with the following identity: Nonmonotonic Logic = Syntax + Nonmonotonic Semantics. The very name 'Nonmonotonic Logic' conveys in such cases similarity with ordinary monotonic logical systems, for which the above equality is clearly appropriate. The analogy is clear, but unfortunately misleading. In ordinary logical systems, the semantics determines, of course, the set of logical consequences of a given theory, but also, and most importantly, it provides a semantic interpretation for the syntax itself. In other words, it provides propositions and rules of a syntactic formalism with meaning, and its theories with informational content. By its very design, however, nonmonotonic semantics is defined as a certain subset of the set of possible models, and consequently does not determine, in turn, the meaning of the propositions and rules of the syntax. For example, radically different theories may have the same nonmonotonic semantics. Furthermore, such a difference cannot be viewed as apparent, since it may well be that by adding further rules or facts to both these theories, we obtain new theories that already have different nonmonotonic models.

8


Thus, the above 'shortcut' definition of Nonmonotonic Logic leaves us without an exact or even clear meaning of the source syntax. Default rules do not bear on their heads information about when and how they can be applied. This is the main reason why the relevant knowledge representation methodology in such systems is essentially an art based on accumulated experience. All these shortcomings have become especially vivid in logic programming. 1.3.1

Logic in logic programming

From its very beginning, logic programming was based on the idea that program rules should have both a procedural and declarative (logical) meaning. This dual interpretation was purported to elevate the programming process by basing it on a transparent and systematic logical representation of real world information. A declarative meaning of a definite (Prolog) program rule was commonly taken to be the meaning of the corresponding classical implication, despite the fact that the (unique) minimal model was singled out as determining the set of conclusions (answers to queries) of the program [van Emden and Kowalski, 1976]. Note that already this setting involved a certain discrepancy, since the logical meaning of a program (viewed as a set of implications) is not determined uniquely by the minimal model, but rather by the set of all its models. This double understanding was further challenged with the introduction of negation as failure as a replacement of the classical negation in logic programs. Clark's completion [Clark, 1978] was commonly accepted as giving more adequate interpretations of logic programs and deductive databases. Nevertheless, for some time these developments peacefully coexisted with an opinion that they reflect a purely pragmatic concession, and an ideal solution should still consist in the full use of classical negation in programs and queries (cf. [Shepherdson, 1988]). Things have changed, however, with the discovery that logic programs with negation as failure allow us to represent significant parts of nonmonotonic reasoning. Moreover, general nonmonotonic formalisms inspired a new kind of semantics for logic programs, the stable and answer set semantics [Gelfond and Lifschitz, 1988; Gelfond and Lifschitz, 1991]. These developments have advanced logic programming to the role of a general computational mechanism for knowledge representation and nonmonotonic reasoning (see the overviews [Apt and Bol, 1994;

Introduction

9

Baral and Gelfond, 1994], and especially [Baral, 2003]). The idea of a dual interpretation of logic programs, procedural and declarative, persisted in all these developments. However, the original question 'What is a declarative meaning of a program rule?' has been replaced with the global question 'What is a declarative meaning of a logic program?', and an answer to this latter question has been commonly thought settled by assigning logic programs some nonmonotonic semantics. Of course, there were reasons for this shift, since already the completion of a logic program does not allow us to single out a modular meaning of a program rule. Unfortunately, the above solution to the problem of determining a declarative meaning of logic programs has turned out to be problematic. To begin with, as was rightly noted already in [Shepherdson, 1988], this solution does not allow us to see the written text of a program as its declarative meaning. For example, adding new rules to a program does not necessarily mean that the extended program contains more information. Worse still, quite diverse programs can have the same 'meaning' according to this understanding, witness such programs as {p f- q) and {q /\a\ a\-£ A}. If we will treat propositions of the underlying language as propositional atoms, it should be clear that the classical models of the above classical propositional theory will stand in one-to-one correspondence with certain sets of propositions. Moreover, the following result shows that they will correspond in this sense precisely to the supported theories. Theorem 2.25 If A is a locally finite sequent theory, then the classical models of comp(A) correspond exactly to supported theories of A. Proof. Assume first that a is a model of comp(A), and ua is the set of propositional atoms that hold in a. Then the right-to-left implications corresponding to the equivalences from comp(A) secure that ua is closed with respect to all sequents from A, and therefore it is a theory of A. Moreover, if A 6 ua, then A must contain a sequent a h b, A such that /\(a U -ifr) holds in a. The latter means that a C wa\{A} and b C u^. Consequently, we obtain ua\{A} h TT^, A, which implies that ua\{A} is not a theory of A. Therefore, any model of comp(A) corresponds to a supported theory of A.


33

Assume now that w is a supported theory of A, and au its associated classical interpretation. Since u is closed with respect to A, all right-to-left implications from comp(A) hold in au. Suppose now that some left-toright implication from comp(A) does not hold in au. Then there exists A e u such that, for any sequent a \- b from A, either a £. u, or A £ a, or u fl (6\{A}) / 0. But by Lemma 2.11, this is exactly what is required for u\{A} being a theory of A, contrary to our assumption that u is a supported theory. Hence, au is a model of comp(A), • Now we will turn to minimal theories. Recall first that a theory w of a well-founded sequent theory A is minimal if and only if it does not contain a loop / such that u\l is also a theory of A (see Theorem 2.20). Hence, the corresponding definition of completion should be extended to cover also nontrivial loops. In addition, in order to obtain a finitary syntactic description, we will consider only sequent theories that are finitely well-founded. Accordingly, given a finitely well-founded sequent theory A, we will define its loop-completion, lcomp(A), as the deductive theory containing the following classical formulas: • /\a —> \/b, for any sequent a\- b from A; • A ' - > V { A ( ° u ~'(b\1)) I a h & e A k i n b ^ 0 & I Da = 0 } , for a n y l o o p / of A . As can be seen, finite well-foundedness is required to secure the correctness of this definition, since it implies finiteness of both loops and the disjunctions in the formulas of the second kind. Then we have Theorem 2.26 // A is a finitely well-founded sequent theory, then the classical models of lcomp(A) correspond exactly to minimal theories of A. Proof. Assume first that a is a model of lcomp(A), and ua is the corresponding set of propositional atoms. Then the formulas of a first kind in lcomp(A) secure that ua is closed with respect to all sequents from A, and therefore it is a theory of A. Moreover, if / C ua, for some loop /, then the corresponding formula of the second kind implies that A must contain a sequent a h b such that / n a = 0, and f\(a U ~L the complementary literal. The notation -i/, where / is a set of literals, will have a similar meaning.

36


A literal dependency relation determined by A will be defined as a relation a U b) = 0, and /\((a U -ife)\->Z) holds in a. As can be verified, these conditions are equivalent to a C (ua\m) U n and 6 C (ua\m) Un. Consequently, (ua\m) U n \~A («a\»™) U n, and therefore (ua\m) U n is not a theory of A. By Theorem 2.29, ua is the only theory of A, and hence A is a unitary sequent theory. Assume now that A is a unitary sequent theory, u is the only theory of A, and au its associated classical interpretation. Since u is closed with respect to A, all the formulas of the first kind in llcomp(A) hold in au. Suppose that some formula of the second kind does not hold in au. Then, for some loop m U ->n C au, we obtain that, for any a h b G A, either a ^ (ua\m) U n, or b A O \/{/\{a

U -.6) I a, A h b £ A}.

It can be easily verified that, if the literal loops are restricted to singular sets of literals {A} and {~1^4}, then llcomp(A) and acomp(A) are logically equivalent, and hence we obtain Corollary 2.31 Let A be a strongly acyclic and strongly locally finite sequent theory. Then u is the only theory of A if and only if the corresponding classical interpretation au is a model of acomp(A).


39

In Chapter 5, the above completions will be used for a constructive description of exact nonmonotonic semantics. 2.4

Circumscription

It turns out that the framework of Scott consequence relations and their theories allows us to give a concise formal characterization of plain circumscription. In the context of Scott consequence relations, the circumscription principle says that only minimal theories should matter in determining what should be inferred or refuted. Accordingly, for a given Scott consequence relation h, we will define new a consequence relation, called a circumscription of h, that is determined by minimal theories of h: Definition 2.10 Let Tm be the set of all minimal theories of a Scott consequence relation K Then \-qrm will be called the circumscription of h and will be denoted by h c . As can be seen, A \~c holds if and only if A is false in all minimal models of K In this case A is taken to be false according to circumscription. Generally speaking, circumscription extends the set of propositions that are seen as refuted with respect to a consequence relation. Moreover, we will see that circumscription can even be characterized as a maximal extension of this kind that still preserves information that is provable in the source consequence relations. The following theorem gives a syntactic characterization of the circumscribed Scott consequence relation. Theorem 2.32 every A £ a-

a K b holds iff h b',b, for any b' such that h b',A, for

Proof. If a V-c b, then there must exist a minimal theory u of h such that a C u and b C u. Now, if A £ a, then h A, u, since otherwise there must exist a theory that is smaller than u. Since a is finite, this implies that there is a finite set b' C u such that h b', A, for any A £ u. But b is also included in u, and hence Y- b, b'. In the other direction, assume that there is b' such that h b',A, for any A £ u, and ¥ b, b'. The second condition implies that there must exist a minimal theory u disjoint from b U 6'. But the first condition implies then that any such u should include every A £ a. Consequently, aY-c b. •

40


As follows from the above theorem, hc is uniquely determined by sequents without premises ('provable disjunctions') that belong to K Accordingly, two Scott consequence relations that coincide on such sequents will produce the same circumscription. This leads us to the following Definition 2.1 • Two Scott consequence relations will be called minimally equivalent if they have the same premise-free sequents. • Two sequent theories will be called minimally equivalent if their corresponding generated Scott consequence relations are minimally equivalent. As the next lemma shows, minimal equivalence amounts to coincidence with respect to minimal theories. Lemma 2.33 Two consequence relations (or sequent theories) are minimally equivalent iff they have the same minimal theories. Proof. Follows from the fact that V b holds if and only if b is disjoint from some minimal theory of h. D The next result states the basic properties of the circumscribed consequence relation. The proofs are immediate. Lemma 2.34 (i) (ii) (Hi) (iv)

For any consequence relation \~,

hCK; (- is minimally equivalent to hc; \~ is minimally equivalent to hi iff\-c=\-^; \~c is a greatest consequence relation among minimal equivalents of\~.

Thus, any equivalence class with respect to minimal equivalence has a greatest element which is also a circumscription of all consequence relations in the class. In addition, the fact that any consequence relation is minimally equivalent to its circumscription means that circumscription does not change the set of premise-free sequents. 2.4.1

Partial deduction

Clearly, finding circumscription of a sequent theory amounts tofindingthe corresponding derived set of premise-free sequents. An important algorithm for determining this set is based on the principle of partial deduction that can be formulated as follows:


41

Replace a sequent A,a\- c of a sequent theory A with the set {a, a; h c, Ci | a,- h c,-, ^4 G A}. As can be seen, partial deduction allows to eliminate propositions from the premises of sequents, and hence it can be employed for obtaining the derived set of premise-free rules. The adequacy of this principle is based on the following result stating that partial deduction preserves minimal equivalence. Theorem 2.35 // a sequent theory A' is obtained from A by an application of partial deduction, then A is minimally equivalent to A'. Proof. Let us assume that A' has been obtained from A by replacing a sequent A, ao h Co with a set of sequents a 0 ,Oj h c o ,Cj,

(*)

for every sequent a; h c,-, A from A containing A among its conclusions. To begin with, any sequent from (*) can be obtained from A by the Cut rule. Consequently, \~&>C.\~&, and hence any theory of A will be a theory of A'. Suppose now that u is a minimal theory of A' that is not a theory of A. Then u is not closed with respect to the sequent A,ao h Co, which means that {A}Uao C u and Co C u. Since u is closed with respect to the sequents from (*), we have also that, for any sequent a,- h c,-, A from A, if a,- C u, then Ci D u ^ 0. But then it can be easily seen that u \ {A} is a theory of A, since it is closed with respect to all its sequents. Consequently, u \ {A} will be a theory of A', contrary to our assumption that u is a minimal theory of A'. Hence any minimal theory of A' will also be a (minimal) theory of A, which shows that A and A' are minimally equivalent. • In the next chapter, this result will be extended to biconsequence relations.

2.4.2

Compact

circumscription

Since circumscription is an extension of the source consequence relation, any theory of the circumscribed consequence relation will also be a theory of the original consequence relation. It turns out, however, that the set of all minimal theories of a Scott consequence relation is not always compact (see Definition 2.2). This means that in some cases the circumscription of

42


a consequence relation h will have theories that are not minimal theories of K Example 2.1 Consider a Scott consequence relation h generated by the following sequents: {h Ai,C} U {H Ai,Bi], for any i > 0. Then u = {A, | i > 0} will be a minimal theory of h, but hc will have also u U {C} as a theory. This unpleasant fact is important, since, as we will see, some useful features of circumscription will only hold under the condition that the set of minimal theories is compact. Somewhat abusing our terminology, we will say that a circumscription of a Scott consequence relation t- is compact, if any theory of hc is a minimal theory of K Definition 2.11 A Scott consequence relation will be called min-finite, if it has only a finite number of minimal premise-free sequents. The following lemma shows that min-finiteness is a weaker property than finiteness. Lemma 2.36 // A is a sequent theory such that [j{b | a (- b 6 A} is finite, then I~A is min-finite. Proof. Let h — [J{b \ a h b 6 A}. If u is a theory of HA, then by Lemma 2.11 u n h will also be a theory of HA- NOW if KA bnh, for some 6, then there exists a theory u that is disjoint from b D h. But then u D h is a theory that is disjoint from b, and therefore HA b. Consequently, if I~A b, then I~A b PI h, and therefore there exists only a finite number of minimal premise-free sequents belonging to HA• Now, the following important result shows that min-finiteness is a sufficient condition for the compactness of circumscription. Theorem 2.37 / / 1 - is a min-finite Scott consequence relation, then it has a compact circumscription. Proof. For any proposition A, we will denote by AA a (finite) set of all minimal sequents of the form \- A,b from K In addition, if Y- A, we will denote by A^ the set of all minimal sequents of the form A, a \- such that a n b ^ 0, for any sequent h A, b from AA (note that in this case b is always non-empty). Finally, we will denote by A the union of all AA and Ax, where A ranges over propositions of the language. We are going to show now that theories of HA coincide with the minimal theories of K


43

It is easy to show that a set u is closed with respect to all AA if and only if Y- u. Consequently, there exists a theory u0 of h such that u0 C u. In addition, u is closed with respect to all A^ iff, for any A, either A g u, or there is a minimal sequent h 6, A such that 6 C u. Consequently, if A G u, then 1- u, A by Monotonicity. Therefore, there can be no models u' of h such that u' is a proper subset of u. Combining this with the earlier condition, we obtain that u is a minimal theory of K Assume now that there is a minimal theory u of h that is not closed with respect to some sequent from A^. Then A £ u and consequently h u, A, since u is minimal. Therefore there must exist a minimal sequent \- b,A such that 6 C u. But then we have a ^ w, for any sequent A, a h from 5^, contrary to the assumption that u is not closed with respect to A^. Consequently, all minimal models of h are closed with respect to all the sequents determining I-A, and hence are theories of the latter. Now, since HA is determined by the minimal theories of (-, it coincides with the circumscription of (-. Therefore, the theories of I-A coincide with theories of hc, and we are done. • The proof of the above theorem contains, in fact, a procedure for constructing a circumscription of a Scott consequence relation in the finite case. To this end we must find first all minimal premise-free sequents that belong to h and then add to them all minimal sequents of the form a h ('refutable conjunctions') that do not change the former set of sequents. Min-finiteness, though sufficient, is by no means a necessary condition for compactness of circumscription. Example 2.2 Consider a Scott consequence relation generated by a set of sequents {\~ A,Bi}, for any i > 0. Clearly, this consequence relation is not min-finite. It has, however, only two minimal models, namely {A} and {B{ | i > 0}, and hence its circumscription is compact. We will show now that a compact circumscription is always generated by adding sequents of the form o h to the source Scott consequence relation. This result shows clearly that the main effect of circumscribing a consequence relation consists in enlarging the set of'refutable conjunctions'. Lemma 2.38 If \-c is compact, it coincides with the least Scott consequence relation containing all sequents of the form a h and h b from h c . Proof. Let h* be the least Scott consequence relation containing all sequents of the form a h and h b from hc. Let u be a theory of h*. Assume that u h c . Then there exists a finite set a C u such that a f~c, contrary to

44


the fact that u is closed with respect to all rules a h from h c . Therefore u V-c, and hence u is included in some theory v\ of h c . Similarly, it can be shown that Yc u, and hence u includes some theory i>2 of h c . However, due to compactness, both v\ and V2 are minimal models of I-, and hence Vi — V2 = u. Therefore, all theories of \~* are theories of h c , and hence h°Cl-*. But the reverse inclusion is obvious, and consequently \~* coincides D with K . Note that, due to the fact that h and h c are minimally equivalent, the set of sequents of the form h b in the formulation of the above theorem is actually the set of such sequents that belong to the source consequence relation h. Unfortunately, the above result does not hold for arbitrary Scott consequence relations, as is shown by the following example of a non-compact circumscription. Example 2.3 Let us consider a Scott consequence relation generated by the following set of sequents (for all natural i): h Ai,B{

Bi h Bi+1

BihC

Bt\- D

Minimal theories of this consequence relation are {A\,..., A,,...} and {C, D, A%,..., An, Bn+i, Bn+2, • • •}, for each natural n. However, its cirIn adcumscription has also a non-minimal theory {C, D,A\,...,Ai,...}. dition, the circumscription contains sequents C h c D and D h c C that are not deducible from the set of constraints added to K

2.5

Supraclassicality

In this last section, we will describe a general class of consequence relations that subsume classical inference. Such supraclassical consequence relations will turn out to be suitable for describing the logical basis of traditional nonmonotonic formalisms such as default and modal nonmonotonic logics. We will consider below consequence relations that are based on a classical language containing the ordinary classical connectives {V, A,-•,->}. As usual, 1= will denote the classical entailment relation with respect to these connectives, and Th its associated classical derivability operator. Definition 2.12 A Tarski consequence relation h in a classical language will be called supraclassical if it subsumes classical inference, that is, (= C K


45

Supraclassicality is equivalent to the requirement that all theories of a consequence relation are deductively closed sets. Lemma 2.39 A Tarski consequence relation is supraclassical iff all its theories are deductively closed sets. Supraclassicality allows for replacement of classically equivalent formulas in premises and conclusions of the rules. In addition, it allows to replace sets of premises by their classical conjunctions: a \- A will be equivalent to f\a h A. Accordingly, supraclassical Tarski consequence relations can be reduced to certain binary relations on the set of classical propositions, namely to relations satisfying, for instance, the following postulates (see [Bochman, 2001]): Dominance If A t= B, then A\- B. Transitivity If A \- B and B h C, then Ah C. And If A \~ B and A h C, then A h B A C. By a conditional theory we will mean in what follows an arbitrary set A of rules of the form A h B, where A, B are classical propositions. Slightly abusing our earlier terminology, for a conditional theory A, we will denote by I~A the least supraclassical Tarski consequence relation containing A, while Cn/\ will denote its associated provability operator. It can be easily verified that theories of h^ are precisely deductively closed sets that are closed also with respect to the rules from A. The following lemma gives a convenient alternative description of this consequence relation. Lemma 2.40 / / H A *S a least supraclassical Tarski consequence relation containing a conditional theory A, then, for any set u of propositions, and any A, u HA A holds iff A is derivable from u by the rules from A and the classical entailment. It should be noted that provable equivalence with respect to a supraclassical consequence relation does not imply that the corresponding formulas are interchangeable in all contexts. For example, even if A f- B and B h A hold, this does not imply, in general, ->A I 'B or AVC h B\lC. This property will hold, however, for 'fully' classical consequence relations described below. Definition 2.13 A Tarski consequence relation will be called classical if it is supraclassical and satisfies the deduction theorem: Deduction If a, A h B, then a r- A -» B.

46


Classical consequence relations already satisfy all the familiar rules of classical inference, such as Contraposition and Disjunction in the Antecedent. Moreover, each of the latter rules is sufficient for classicality. Note that even classical consequence relations still do not coincide, in general, with the classical entailment l=; the latter can be described as the least classical (or, equivalently, least supraclassical) consequence relation. There is, however, an intimate connection between the classical entailment and classical consequence relations. Note that any rule A h B of a classical consequence relation is equivalent to \- A —> B. Consequently, any classical consequence relation can be seen as a classical entailment with some set of propositions added as additional, nonlogical axioms. And in this respect arbitrary supraclassical consequence relations allow additional freedom in that they permit the use of auxiliary nonlogical inference rules A h B that are not reducible to the corresponding material implications. The following definition provides a two-step extension of supraclassicality to Scott consequence relations. Definition 2.14 A Scott consequence relation will be called weakly supraclassical, if it satisfies: S u p r a c l a s s i c a l i t y If a 1= A, then a\~ A.

and (strongly) supraclassical, if it satisfies, in addition, Falsity

fK

As can be seen, a Scott consequence relation is weakly supraclassical if and only if its Tarski subrelation is supraclassical. And just as for the latter, this requires all theories of a Scott consequence relation to be deductively closed. Strong supraclassicality amounts, in addition, to exclusion of the inconsistent theory Pr from consideration. In the supraclassical case, such a model does not have a useful meaning (at least for our study), precisely because it is classically inconsistent. Accordingly, a Scott consequence relation is supraclassical if and only if all its theories are consistent deductively closed sets. Just as for Tarski consequence relations, supraclassicality allows for replacement of classically equivalent formulas in premises and conclusions of sequents, as well as replacement of sets of premises by their classical conjunctions: a h b will be equivalent to f\a h b. Multiple conclusions, however, cannot be replaced in this way by their classical disjunctions. Taking the simplest case, a sequent h B, C is not reducible t o h 5 V C ; the


47

latter says that any theory should contain BV C, while the former asserts a stronger constraint that any theory should contain either B or C. Finally, in order to obtain a classical sequent calculus, we need only to add the following postulate: Completeness

h A, ->A.

Let us say that a Scott consequence relation is classical, if it is supraclassical and satisfies Completeness. Any theory of a classical Scott consequence relation will contain either A, or ->A, for any proposition A, and hence it will be a world. Consequently, such a relation will satisfy already all the postulates of a sequent calculus. In particular, we will have that any sequent a h b of such a consequence relation will be reducible to /\ a \- \J b, so it will be equivalent also to a classical Tarski consequence relation. Completeness is equivalent to each of the following rules: (Disjunction^ A V B \- A, B. ( D i s j u n c t i o n ) If A, a h b and B, a h b, then A\/ B,ah b. As a general agreement for what follows, for a sequent theory A in a classical language, we will denote by A the set of classical implications corresponding to the sequents from A:

~£ = {/\a->\/b\a\-b<E A}. Then it can be easily verified that a proposition A is provable in the least classical consequence relation containing A if and only if A 1= A. 2.5.1

Minimal and classically supported theories

The definition of a minimal theory of a consequence relation can be directly extended to supraclassical relations. Moreover, its syntactic description (cf. Lemma 2.18) for the (strongly) supraclassical case can now be simplified by excluding the case of Pr: Lemma 2.41 A set u of propositions is a minimal theory of a supraclassical Scott consequence relation if and only if u = {A | I- u, A}. The definition of a supported (pointwise-minimal) theory, however, was essentially syntactic, and hence it cannot be transferred directly to the supraclassical case due to the requirement of deductive closure. Still, a proper modification of this notion can be obtained as follows.

48


In the structural setting of general Scott consequence relations, a minimal change of a theory (= set of propositions) u has been achieved simply by a removal of a single proposition. In a deductively closed setting, however, the corresponding change can be defined as a transition from a deductively closed theory to some its maximal proper sub-theory. Note that, for any deductive theory «, there always exist maximal deductive theories that are properly included in u. The set of all such subtheories (plus u itself) is usually denoted by wJ_; it plays an important role in the AGM theory of belief change (see [Alchourron et a/., 1985; Bochman, 2001]). A similarity with the notion of pointwise minimality can be established if we identify deductively closed sets of propositions with the associated sets of worlds. Then every maximal sub-theory of a given theory u can be obtained by adding a single new world to the set of worlds corresponding to u. In other words, these theories from M_L can be obtained as sets of the form ufla, where a is a world that does not include u (this is precisely the Grove connection [Grove, 1988]). The following notion of a classically supported theory embodies this understanding of minimality. Definition 2.15 A theory u of a supraclassical Scott consequence relation h will be called classically supported if u is the only theory of h in u_L. Clearly, any minimal theory of a supraclassical consequence relation will be classically supported, though not vice versa. The next lemma provides a syntactic description of classically supported theories. Lemma 2.42 A theory u of a supraclassical Scott consequence relation h is classically supported if and only if a fl u h a n u, for any world a that does not include u. Proof. The above condition says, in effect, that no set of the form uf\a is a theory of K Due to the Grove connection, this means that u is classically supported. • The above description of supported theories can be simplified by using relativized consequence relations. Note that if h is a supraclassical consequence relation, and u is a deductively closed set, then the relativized consequence relation h u , as defined earlier, will also be a supraclassical consequence relation. For a deductively closed set u, let us denote by hjf the least classical consequence relation containing h u . Then we have


49

Corollary 2.43 A theory u of a consequence relation h is classically supported if and only if h£ A, for any Ae u. Proof. Note first that theories of h" are precisely theories of h" that are worlds. Consequently, the above condition says that u C a, for any world a that is a theory of h u , that is, a ¥u a. By Lemma 2.16, a is a theory of l-u if and only if a D u is a theory of K Hence the result follows from the preceding lemma. • The above description can be transformed into a relatively simple algorithm of checking whether a theory is classically supported. Recall that A is provable in the least classical consequence relation containing A if and only if A t= A. Consequently, we obtain Corollary 2.44 A theory u of a sequent theory A is classically supported if and only if Au t= A, for any i £ u . Finally, we will give two sufficient conditions for coincidence of minimal and classically supported theories of a sequent theory. These conditions will play an important role in Chapter 9. Definition 2.16

A sequent theory in a classical language will be called

• premise-free, if it contains only sequents of the form f- a; • literal, if it contains only sequents of the form I h m, where /, m are sets of classical literals. Theorem 2.45 Classically supported theories of a premise-free or literal sequent theory coincide with its minimal theories. Proof. Assume first that A is a premise-free sequent theory, and suppose that u is a non-minimal theory of A, that is, there is a theory v of A such that v C u. Let a be an arbitrary world that includes v, but does not include u. Then u PI a will also be closed with respect to all the sequents from A, and hence u will not be classically supported. Indeed, if 1- a £ A, then A £ v, for some A E a (since v is a theory). Consequently A £ a n u, and hence a f\ (a f\ u) ^ 0. Assume now that A is a literal sequent theory, and let a be a world that contains all literals from v, negations of all literals from u\v, and u for

If\\-C'r is a circumscription an

V assumption

o/lh, then a : b \\-clr c : d holds

theory v o/ lh such that b C v and d

Cv.

Proof, a : b lh ctr c : d does not hold iff there exists a positively minimal bitheory (u,v) of lh such that aCu, bCv, cCu and d C v. As can be seen, in this case v is an assumption theory of lh and, moreover, u is a minimal theory of lh|J\ Consequently, the latter condition amounts to the fact that o h j c does not hold. Hence the result. • As a consequence of the above result, the basic facts about circumscription of biconsequence relations, stated below, are completely analogous to the corresponding facts for Scott consequence relations, including their proofs. The following corollary from the above lemma provides a direct syntactic description of the circumscribed biconsequence relation. The proof is immediate from Theorem 2.32.

Corollary 3.57 a:b \\-cir c:d holds iff :b,b' lh c,c':d,d', for any b',c',d' such that :b' lh c',A:d', for every A £a. As can be seen from this description, the circumscription is uniquely determined by the autoepistemic bisequents belonging to a biconsequence relation. Practically all the results about circumscription of Scott consequence relations are extendable to biconsequence relations. This pertains also to compact circumscriptions, for which any theory of lhcir is a positively minimal theory of lh. The following results are straightforward extensions of the corresponding results for Scott consequence relations, stated in Chapter 2. Definition 3.33 A bisequent theory will be called min-finite, if, for any proposition A, it contains no more than a finite number of minimal bisequents of the form : 6 lh c, A : d. Theorem 3.58 //lh is a min-finite biconsequence relation, then it has a compact circumscription.

84


Moreover, just as for Scott consequence relations, for compact circumscription only bisequents of the 'constraint' type a : b Ih : d need to be added to I h in order to obtain \\-clr. L e m m a 3.59 If \Vc%r is compact, it is the least biconsequence relation containing all bisequents of the form a : 6 Ih : d and : b Ih c : d from lh C!r . As before, this result does not hold for arbitrary biconsequence relations. A suitable counterexample can be easily obtained from the corresponding example given earlier for Scott consequence relations. The above lemma suggests a general method of circumscribing a biconsequence relation. To this end we must find first all minimal autoepistemic bisequents that belong to Ih, and then add to them all (minimal) bisequents of the form a :b\Y : d that do not change the former set of bisequents. For default biconsequence relations, the above characterization of circumscription can be strengthened as follows: L e m m a 3.60 / / lhC!r is a compact circumscription of a default biconsequence relation, then it is the least biconsequence relation containing all bisequents of the form a : Ih : d and : b Ih c : from lh c "\ Proof. Let Ih* be the least biconsequence relation containing all sequents of the form a :lh : d and : 6 Ih c : from lh" r . Clearly, lh*ClhC!>. Let (u,v) be a bitheory of IK. Then -.vlP* u :. But :bVf*c: holds only if : bVfc%r c :. Consequently, : v Vfc%r u :, and therefore there exists a bitheory («i,«i) of lhc"* such that t i i C u and v\ C v. Similarly, u :Jf*: v implies u :l^ctr : v, and therefore there is a bitheory (u2>«2) of lh clr such that u C ui and v C V2- Now, since Ih is a default biconsequence relation, and (wi,î) is a positively minimal bitheory of Ih, both (ui,v) and (ui,t>2) are bitheories of Ih. But (u2,V2) is also positively minimal in Ih, and consequently U\ = «2 = y- Moreover, (it, v) is positively minimal in Ih, since otherwise (u2, V2) would not be positively minimal. Thus, any bitheory of Ih* is a bitheory of \\-clr, and therefore \\-"r coincides with Ih*. ' • The above strengthening considerably reduces the 'search space' for constructing circumscription by reducing it to bisequents of the form a : Ih : d. Note, however, that the circumscription of a default biconsequence relation need not be a default biconsequence relation by itself.

Chapter 4

Four-Valued Logics

This chapter can be seen as a general introduction to a theory of fourvalued reasoning based on the formalism of biconsequence relations. The latter formalism has an important advantage in this respect in that it does not depend on a particular choice of four-valued connectives. Moreover, we will show that any such connective is definable in it via introduction and elimination rules as in the classical sequent calculus, with the only distinction that we will have a pair of introduction rules and a pair of elimination rules corresponding, respectively, to the two contexts forming a four-valued interpretation. In this way we will obtain also formal representations for a number of important four-valued logics that will be used in the sequel.

4.1

Introducing Connectives

Due to the correspondence between four-valued interpretations and bimodels, any four-valued connective #(Ai,..., An) can always be determined by a pair of conditions describing, respectively, when it is true and when it is false. Consequently, it can be described by a pair of definitions: v)=#{A1,...,An) = F+[A1,...,An] u^#(A1,...,An) = T-[A1,...,An] where F+[Ai,..., An] and T~ [A\,..., An] are classical logical formulas in the meta-language generated by elementary propositions of the form v f= At and v=\A{. We will show that introduction and elimination rules for such a connec85

86


tive can be always given in the following form: + lff

j

, lff

j

l #

j

l

j

*

{a, a,- : b, b,- Ih c, c,- : d, d,} ( ! < » ' < &i) a,#(A1)...,4n):6ll-c:d {a, a,- : 6, 6,- Ih c, c,- : d, dj} (1 < i < fe2) ^ b l h c , ^ , . . . , ^ ) ^ {a, a,- : b, 6,- Ih c, c; : d, d,} ( ! < » ' < fa) a:b\\-c:d,#(A1,...,An) (1 < z < fc4) {a, a,- : 6, bt Ih c, Cj : d, d,} a:b,#(A1,...,An)lhc:d

where a,, b,-, c,- and d; are subsets of { A i , . . . , An}. The following theorem shows that any four-valued connective can be characterized by such rules added to a biconsequence relation. This theorem can be seen as a generic Completeness Theorem for biconsequence relations in languages containing four-valued connectives. Theorem 4.1 Let # ( ^ 4 i , . . . , An) be a four-valued connective determined by D^. Then there are four rules of the above form such that any biconsequence relation satisfying these rules is generated by a set of four-valued interpretations satisfying D#. Proof. Let us assume that T+\A\,..., An] is represented in a disjunctive normal form C\ V • • • V C%, where each C{ is a conjunction of 'literals' of the form v\=Aj, v\fcAj, u=\Aj, or i/^\Aj. Then we will introduce a rule of the form # £ ' + such that Aj belongs to a; (respectively, to b,-, c,-, or dj) iff

v \=Aj (v=\Aj, v\£Aj, or i/^\Aj) belongs to C*. Assume now that T+[Ai,..., An] is transformed into a conjunctive normal form T)\ A • • • AVm, where each X>; is a disjunction of the same 'literals'. Then we will introduce a rule # / + such that Aj belongs to a,- (respectively, to b,-, a, or dj) iff v ^ Aj (v ^j Aj, v (= Aj, or v =| Aj) belongs to X\-. In the same way, a disjunctive normal form oiT~ \A\,..., An] generates a rule of the form # / ~ , while its conjunctive normal form generates a rule of the form #E~. Assume that Ih is a biconsequence relation satisfying the above rules, (u, v) is some its bitheory and V(u,v) a four-valued interpretation correspondAn) belongs to u, at least ing to (M, V). Then #E+ implies that if #(A\,..., .,An] one of the conjuncts C,- of a disjunctive normal form of T+\A\,.. should be such that a,- C u, b{ C v, C{ C u and d,- C v. Consequently, "(«, A ) implies J " + [ ^ i , . . . , An] for 1/ = V(u,v)- Similarly,

87

Four-Valued Logics

# / + implies that if V(u,v) W #(-îi • • • > ^n)> o n e °f the disjuncts of a confor v = ^(u,,,) should be false, junctive normal form of T+[Ai,...,An\ and hence T$\A\,..., An] itself is false with respect to this interpretation. Thus, U(Utv) satisfies the first condition of -D#. In the same way it can be shown that the other two rules imply the validity of the second condition from Z)# for V(u,v)- Consequently, all canonical interpretations of Ih satisfy D#. Now the result follows from the Representation Theorem for biconsequence relations, since any biconsequence relation is generated by its canonical semantics. • An implementation of the procedure given in the proof of the above theorem to a particular class of classical four-valued connectives will be presented below. Just as for the classical sequent calculus, the rules corresponding to four-valued connectives satisfy the subformula property, and hence allow us to reduce any bisequent involving such connectives to a set of bisequents containing atomic propositions only, without the use of the two Cut rules. This is simply a syntactic expression of the fact that the value of any proposition involving only truth-functional connectives in an interpretation is uniquely determined by the values of its atomic propositions. Bisequents that involve only atomic propositions will be called basic ones. Thus, for any given language containing four-valued connectives, there is a one-to-one correspondence between biconsequence relations and their restrictions to the basic bisequents. Note that the latter can be considered as biconsequence relations in their own right, namely as biconsequence relations in the language without connectives. Such biconsequence relations will also be called basic. Thus, any biconsequence relation involving only four-valued connectives is equivalent to some basic biconsequence relation. Finally, we will briefly describe yet another general way of characterizing four-valued connectives in biconsequence relations, namely by a set of bisequents having one of the forms: (#£+)

a,#(A 1 ,..., J 4 n ):6lhc:d

(#/+)

a:b IH c,#(Au

(#£ 0 ~)

a:b\\-c:d,#{Au...,An)

(#J 0 ")

a:b, #(AU ..., An) Ih c:d,

..., An):d

where a, b, c and d are subsets of {Ai,..., An}. This characterization is actually a four-valued generalization of the corresponding description of classical connectives in the framework of Scott consequence relations given in [Gabbay, 1981; Segerberg, 1982].

88


The next theorem shows that any four-valued connective can be characterized in this way. Theorem 4.2 Let # ( ^ 4 i , . . . , An) be a four-valued connective determined by D#. Then there are rules of the above form such that any biconsequence relation satisfying these rules is generated by a set of four-valued interpretations satisfying D^. Proof. As in the proof of the preceding theorem, let us assume first that T$[A\,..., An] is represented in a disjunctive normal form C\ V • • • V C%, where each C, is a conjunction of literals of the form i/\=Aj, v\{^ Aj, v=\ Aj, or v^\ Aj. Then, for every Ci, we will introduce a bisequent of the form #/^~ such that Aj belongs to a (respectively, to 6, c, or d) iff v f= Aj {y=\ Aj, v y=. Aj, or v j \ Aj) belongs to Ci. Assume now that !F+[A\,..., An] is transformed into a conjunctive normal form T>i A • • • A Vm. Then, for every 75,-, we will introduce a bisequent of the form #EQ such that Aj belongs to a (respectively, to 6, c, or d) iff vtf=Aj (v^\Aj, v\= Aj, or v =j Aj) belongs to 1>i. In the same way, a disjunctive normal form oiT~ \A\,..., An] generates bisequents of the form #EQ , while its conjunctive normal form generates bisequents of the form # / ^ . Assume now that Ih is a biconsequence relation satisfying the above rules, and V(u,v) ls a four-valued interpretation corresponding to some bitheory (u, v). Then each bisequent #JEO~ implies that if V(u,v) t= # ( ^ i ) • • • > An), then the corresponding disjunct Z>; of a conjunctive normal form of F£[A\,..., An] is true for v = V(u,v). Consequently, all such bisequents imply jointly that F+[Ai,..., An] is true for v = V(u,v)- Similarly, all bisequents of the form # / + jointly imply that if V(u,v) t ^ # ( ^ i i • • -i An), T+[A±,..., An] should be false for v = vû>vy Thus, V(u,v) satisfies the first condition of Z?#. In the same way it can be shown that the bisequents of the other two kinds imply the validity of the second condition from D # . Consequently, all canonical interpretations of Ih satisfy Z?#. Now the result follows from the Representation Theorem for biconsequence relations. •

4.1.1

Classical four-valued

connectives

A particular class of four-valued functions turns out to be of special interest in our intended use of biconsequence relations. Since we will be primarily interested in what information a bi-context (four-valued) reasoning can give us about ordinary, classical truth and falsity, that is, about t and f,

89

Four- Valued Logics

we should require that the corresponding four-valued reasoning must agree with the classical one in cases when the context does not involve inconsistent or incomplete information. To secure this requirement, we should restrict our attention to connectives that are classical in the sense that they are conservative on the subset {t,f} of classical truth-values. In other words, such connectives should give classical truth-values when their arguments receive classical values t or f. The set of classical connectives is actually well-known in the literature on many-valued logics, and it can be described in a number of ways. A most important 'global' property of this set is that it is functionally pre-complete in the class of all four-valued functions (see [Hendry, 1983]): adding an arbitrary non-classical function to this set allows to express any four-valued connective whatsoever. It turns out that there are four natural and mutually independent connectives that are jointly sufficient for defining all classical four-valued functions. The first is the well-known conjunction connective: v\=A/\B vÂf\B

iff \K

v \= A and is \= B vÂo*v=\B.

A conjunction of two propositions is true when both its conjuncts are true, and false when at least one of them is false. Thus, such a connective gives us a nearest four-valued counterpart of the classical conjunction. Next, there are two unary connectives that can be seen as two alternative extensions of classical negation to the four-valued setting: v\=-Â iff vÂ v\=~A

iff v=\A

v=\Â iff vj\A

{D^)

i/=|~A iff

(D ~)

v\=A.

Note that these are the only connectives that coincide with the classical negation on the classical truth-values and satisfy the Double Negation rule. The difference between the two is that the second one 'switches' the contexts between truth and falsity, while the first one retains the context. Accordingly, we will call -i and ~ a local and global negation, respectively. Note also that each of them can be used together with the conjunction to define a natural disjunction connective: Aw B = ~ ( ~ A A ~ 5 ) = -.(-vlA-.B). As could be expected, a disjunction is true when at least one of the disjuncts is true, and false when both disjuncts are false.

90


Finally, the following unary connective A can be seen as a kind of a modal operator. It determines a rudimentary modal logic definable in the four-valued setting (and becomes trivial in the classical context). v\=AA

iff

vfiA

vÂA

iff

v=\A.

(DA)

The operator A is closely related to the modal operator that will be used in a modal extension of our formalism (see Chapter 10). We will use in what follows also alternative unary connectives L and — defined below: v\=LA iff

u\=A

v=\LA iff

vfcA

iff

v=\A

v^\-A

iff

vj\A

v\=-A

Given the global negation ~, all these connectives are interdefmable: L = — = ~A~

- = ~A = L~

A = ~L~ = — = —

The connective — is similar to Heyting's intuitionistic negation, and is related to the negation of the here-and-there logic (see [Pearce, 1997]). Accordingly, it will be called an ht-negation in what follows. As can be verified, — satisfies the de Morgan rules. It does not satisfy, however, the Double Negation rule, only a weaker 'Triple Negation' rule A = —A. Even for the classical logic, the choice of a natural functionally complete set of 'basic' connectives is not unique. We have still less reasons for reaching agreement about what could be seen as a natural functionally complete set of classical four-valued functions. Nevertheless, the main advantage of the suggested choice of the basic connectives for our study is that it is modular in the sense that a number of important subclasses of four-valued connectives, discussed below, are obtained simply by removing some of the basic connectives. The following proposition shows that any classical four-valued function is representable using our four basic connectives. Theorem 4.3 The set {A, -i, ~, A} is functionally complete for the set of all classical four-valued functions. Proof. Let v be a 4-interpretation restricted to atomic propositions pi,..., pn. We will define a proposition Av corresponding to v as follows: Au

=

PiApiA---ApnApn,

91

Four-Valued Logics

where pi is either p,- or -A,A:\\-

\\-A,- j \ Ap iff v^\ AuFnvF or ;v $ AvF\uF iff i/ G VF- Thus, Af determines the same four-valued function as F. • The class of conservative connectives will turn out to be particularly useful for describing three-valued reducts of our four-valued formalism. There exists a useful replacement for our representative set of conservative connectives (see [Arieli and Avron, 1996]), which consists in replacing the operator A with two connectives: the syntactic falsity constant f and the implication connective D defined as follows: vÂDB iff vÂorv^B v =| A D B iff v |= A and v =| B. The above connective appears to be a nearest counterpart of material implication in the four-valued context. The following identities show that A is interdefinable with {f, D} in our present (conservative) setting. A D B = A~A V B

f = (AA A ~AA)

AA = ~A D f.

* ** At this point it becomes easy to verify that all the connectives in the set {A, ->, ~, A} are independent. Indeed, the conjunction is independent from the rest, since it is the only binary connective in the list, the local negation -i is independent, since it is the only non-conservative connective, the operator A is the only non-invariant connective, while the global negation ~ is independent, since the rest of the connectives are positively local: the

Four- Valued Logics

97

truth of a compound formula is determined only by the truth valuation on its components. 4.1.5

Normal forms

Just as in classical logic, it is convenient sometimes to reduce a logical formula in a four-valued language to an equivalent formula in a certain normal form. Fortunately, it turns out that in our four-valued case any formula can also be reduced to either a conjunctive or disjunctive normal form with respect to the local classical connectives, that is, to a conjunction of disjunctions (respectively, a disjunction of conjunctions) of literals. There are, however, two essential differences. First, we have a richer collection of literals. Second, the set of possible literals sufficient for a reduction will not be unique. Definition 4.2 Four-valued propositions A and B will be called 4equivalent (notation A = B) if they receive the same values in any fourvalued interpretation. Propositions A and B are 4-equivalent if and only if, for any four-valued interpretation, v\=A iff v j= B and v=Â\Ri>=\B. It is easy to show that 4-equivalent formulas are interchangeable in any larger context without changing the truth-value of a compound formula, or the validity of bisequents. In this sense, 4-equivalence constitutes a natural counterpart of classical logical equivalence. As in the classical case, the reducibility of four-valued formulas to a disjunctive or a conjunctive normal form is based on three kinds of rules. First, we have the following distributivity equivalences: AA(BVC) = {AAB)V{AAC) i V ( B A C ) = (AVB)A(iVC) Second, all the unary connectives, described earlier in this chapter, can be distributed over disjunction and conjunction. Thus, for every negative connective xiG {->, ~, —} we have de Morgan laws: cxi (A A B) = xi AM M B

xi(4VB) = M i A i x B ,

while for every positive connective \x£ {L, *, A} we have a simple distribution: ix (A A B) = xi A/\

M

B

xi (A V B) = ix AV ix B.

98


As a result, any formula in our language is reducible to a conjunction of disjunctions (or a disjunction of conjunctions) of 'quasi-literals', that is, atoms with unary connectives nested in front of them. The following sets of equivalences will show, however, that such nested connectives will always be reducible to single ones. First, we have that the local negation commutes with every positive unary connective {L, A,*}: -iA = A~i = —

-i* = *-i = ~

-iL = L~i.

As a consequence, the local negation can be 'moved out', and therefore any chain of nested unary connectives is reducible to a chain of the positive connectives, possibly preceded with the local negation. Furthermore, the following equivalences show that any two consecutive positive connectives are reducible to a single one. **A = A LA = L* = *A = AA = A AL = *L = A* = LL = L. In what follows, by a 4-atom we will mean any proposition of the form p, Lp, *p or Ap, where p is a propositional atom. By a J-literal we will mean, accordingly, a 4-atom or its local negation. Then the above equivalences show, in effect, that any quasi-literal is reducible to some 4-literal. As a result, we obtain Lemma 4.8 Any four-valued proposition is 4-equivalent to a proposition in a conjunctive (resp., disjunctive) normal form built from 4-atoms. It turns out, however, that the above construction still does not give us maximally reduced normal forms. The reason is that any of the three positive unary connectives {L, A, *} is 'clausally' definable in terms of the other two. Namely, we have the following equivalences: LB = (*B A ->AB) V {*B A B) V (-.AB A B) *B = (-.B A AB) V (LB A AB) V (-.B A LB) AB = (B A ->LB) V (B A *B) V (^LB A *B) B = (LB A -.*£) V (LB A AB) V (AB A ->*B). The last equivalence above shows that even 'pure' literals p and -ip are eliminable, if we have the rest of the 4-literals.

Four-Valued Logics

99

Thus, as our final result, we obtain Theorem 4.9 Any four-valued proposition is 4-equivalent to a proposition in a conjunctive (resp., disjunctive) normal form built from 4-atoms of any three kinds from the set {p, Lp, Ap, *p}.

4.1.6

Alternative

representations

Having the four-valued connectives at our disposal, we can transform bisequents into more familiar inference rules, and even to ordinary logical formulas. Recall that -i« denotes the set {-*A | A G u}. The notation ~w, Aw and their combinations will have a similar meaning. The following representation of bisequents can be easily obtained from the characteristic rules for the relevant connectives. Lemma 4.10 following:

Any bisequent a : b Ih c : d is equivalent to each of the

a, ~6 : If- c, ~d :

(1)

II--na,c :• A that will form a basis for production and causal inference relations studied in Chapter 8. A distinctive feature of such rules is that they do not satisfy the Reflexivity postulate for consequence relations. Nevertheless, such inference relations will be shown to constitute a most immediate generalization of classical logic that allows representation of nonmonotonic reasoning. Note also that, due to representations (3) and (5), both the global negation ~ and the ht-negation — are sufficient in this setting for transforming bisequents into four-valued formulas. Finally, the representation (4) shows that the conservative language {A ~, A} is also capable to provide such a formula-based representation. Moreover, the latter representation will also turn out to be suitable for representing three-valued logics. 4.1.7

Classical

translation

Now we will show that biconsequence relations, and four-valued logics in general, can be faithfully translated into classical logic. The translation, described below, exploits the fact that four-valued interpretations correspond to pairs of classical interpretations; the latter will

Four-Valued Logics

101

be 'simulated' by using two different copies of propositional atoms. More precisely, for any propositional atom p, we will introduce a new propositional atom p'. Then, as a first step, we will describe two translations \A\ and ||^4|| of four-valued formulas A to classical formulas in the extended language. The translations will be defined by a simultaneous recursion as follows: \p\=p \AAB\ = \A\A\B\

\\p\\=pf \\A A B\\ = \\A\\ A \\B\\

hA|=HA| |~A| = H|A||

|hA|| = HM| ||~A|| = HA|

\AA\ = \\A\\

||AA|| = |H|.

As can be seen, the above translation steps closely follow the corresponding definitions of the four-valued connectives with respect to the truth and falsity valuations. Moreover, since the above set of connectives is functionally complete in the class of all classical four-valued connectives, any such connective could be readily included in these translations1. For a set u of four-valued propositions, we will denote by \u\ and ||u|| the sets of corresponding translations of all propositions from u. Finally, for a bisequent p = a:b II- c:d in an arbitrary four-valued language, we will denote by [p] the classical formula

/\(Hui|6||)->V(lcluHHI)Similarly, for a bisequent theory A, we will denote by [A] the corresponding set of classical translations. Then we obtain Theorem 4.11 A bisequent p is derivable from a bisequent theory A if and only if [A] 1= [p]. Proof. Note first that, under the above translation, the postulates of biconsequence relations correspond to classically valid propositions and rules; this gives us the direction from left to right. In the other direction, assume that p — a:b II- c:d is not derivable from A. Then there exists a four-valued interpretation v that validates A, but not p. Then we define a classical interpretation i in the extended language as follows: for any propositional atom p, i 1= p = v t= p 1

i 1= p' = v ^) p.

Actually, the translations are easily extendable to arbitrary four-valued connectives.

102


Then, by an obvious induction on the complexity of four-valued formulas, we obtain that, for any formula A, v 1= A if and only if i 1= \A\ z/=j A if and only if i ¥ \\A\\. As an immediate consequence of the above equivalences, we obtain that an arbitrary bisequent p0 is valid in v if and only if the classical formula [po] is valid in i. Consequently, all formulas from [A] are valid in i, while [p] is invalid in i, which shows that [A] ¥• [p]. This completes the proof. • The above result shows that our construction provides a faithful and general embedding of four-valued inference into classical logic. Furthermore, using our previous results, this embedding can be extended to various special kinds of four-valued inference, described earlier. Thus, for consistent, complete, ordered and semi-classical inference, the corresponding classical translations are obtained simply by adding the translations of the corresponding postulates as additional premises in the above classical entailment. In addition, for the regular and invariant inference, the corresponding translations are obtainable using, respectively, Corollary 3.26 and Corollary 3.34. Corollary 4.12 • A bisequent p is derivable from a bisequent theory A with respect to a regular inference if and only if [A], [Ar] 1= [p]. • A bisequent p is derivable from a bisequent theory A with respect to an invariant inference if and only if [A], [A°] 1= [p\. 4.2

Four-valued Entailment and 4-Theories

For a given four-valued language £, we will denote by ||= £ the least biconsequence relation in £. This biconsequence relation will be called the bisequent calculus (for £). Clearly, |(=£ is similar to the classical sequent calculus in that it contains only bisequents that are derivable using the structural postulates of biconsequence relations and the introduction/elimination rules for the connectives from £. In this sense, it represents the pure four-valued logic for the language £. In what follows, we will suppress the index £ in \\=c whenever it will be clear from the context, or else our results will invariably hold for any £.

Four-Valued Logics

103

Now we will introduce a four-valued counterpart of the notion of a deductively closed theory. Definition 4.3 A pair (u, v) of sets of propositions in C will be called a 4-bitheory, if it satisfies the following two conditions, for any proposition A from C: • If u : \\=A:v, then A £ u; • If u : A ||= : v, then A £ v. A 4-bitheory will be called C-consistent, if u : |[£ : v. 4-bitheories can be seen as bitheories that are closed with respect to the four-valued entailment. Similar to the classical case, the following lemma shows that a 4-bitheory can be viewed as an intersection of a set of fourvalued interpretations. Lemma 4.13 A pair («, v) is a (consistent) 4-bitheory if and only if there is a (nonempty) set I of four-valued interpretations such that u = {A \ is \= A, for any v G / }

and v = {A \ v^\A, for any v G / } .

Proof. Let us say that a four-valued interpretation v is a model of (u, v), if v\=B, for any B E u, and i/^\C, for any C £ v. Then u: \\=A:v says that i/\=A, for any model v of (u, v), while u:A ||= :v says that v^\A, for any model of (u, v). This gives us the direction from left to right, when / is the set of all models of (u, v)) (note that (u, v) is consistent iff it has a model). Assume now that (u, v) is determined in the above way by a set of / of interpretations, and u: \\=A:v. Clearly, any interpretation from / will be a model of (w, v), and consequently ^|=A, for any v 6 / . Therefore, A £ u. Similarly, it can be shown that if u: \\= A:v, then A G v. Thus, (u, v) is a 4-bitheory. In addition, if / ^ 0, then it clearly will be consistent. • It turns out that the definitions of many four-valued connectives are also preserved in the setting of 4-bitheories (provided the connectives belong to the language). The proof is straightforward, given the above lemma. Corollary 4.14 • • • •

If (u, v) is a 4-bitheory, then

AAB£uiffA£u and B G u; AAB £v iffAev and B £v; *A G u iff A € v; *A G v iff A G u;

• ~ A G u iff ->A G v;

104


• ~A £ v iff ->A 6 u; and similarly for —, L and A.

Note, however, that, just as for classical deductive theories, the corresponding equivalences do not hold for the local negation -i and disjunction. The above conditions for the conflation * and global negation ~ imply, in particular, that, for the languages containing these connectives, the positive component of a 4-bitheory uniquely determines its negative component, and vice versa: v ~ {A | *A G u}

u = {A | *A G v}.

Accordingly, for languages of this kind, the following more usual notions will turn out to be appropriate. Definition 4.4

Given a four-valued language £,

• A set M of propositions (positively) entails a proposition A (notation u 1=4 A), if u: \[=c A:. • A set u is a (positive) ^-theory, if it is closed with respect to 1=4. The entailment relation (=4 can be seen as a natural extension of classical entailment to four-valued languages. In particular, local classical propositions (that is, propositions in the language {A, -1}) behave in a fully classical way with respect to (=4. The following simple result connects the notion of a 4-theory with the earlier notion of a 4-bitheory. Lemma 4.15 A set u is a positive 4-theory if and only if [u, v) is a 4-bitheory, for some v. Proof. Take v as the set of all propositions A such that v ^j A, for any four-valued interpretation v that validates u. • By the above lemma, 4-theories are precisely the positive components of 4-bitheories. As a consequence, we immediately obtain that any 4-theory is an intersection of positive components of four-valued interpretations. By a 4-world we will mean a maximal consistent 4-theory. Just as for classical logic, 4-worlds can be given the following alternative descriptions: Lemma 4.16

The following conditions are equivalent:

• a is a 4-world; • a = {A I v t= A}, for some four-valued interpretation v.

Four- Valued Logics

105

• a is closed with respect to £4 and saturated: i V B g a only if either A £ a, or B G a. Due to the above saturation property, any 4-world is uniquely determined by 4-atoms that belong to it. The classical translation of four-valued languages, described earlier, extends also to 1=4. Namely, Theorem 4.11 immediately gives Corollary 4.17

For any set u of four-valued propositions, and any A, uP4A

iff

\u\t\A\.

There is a useful consequence of the above equivalence that stems from the fact that validity of local classical propositions in each of the two contexts of a four-valued interpretation do not depend on each other. Lemma 4.18 If u,v are sets of (local) classical propositions, and A,B are classical propositions, then u, *v N4 A V *B iff either u 1= A or v 1= B. As a result, we obtain the following Corollary 4.19

// u, v, A are classical propositions, then u, *v 1=4 A iff either u t= A or v\= u, *v t=4 *A iff either v (= A or u\= .

This result will be used in what follows.

4.3

Invariant Logic

In this section we will give a detailed description of four-valued logic in the invariant language. This logic will be used later in this study for describing partial generalizations of nonmonotonic semantics.

4.3.1

Axiomatic representation

As we have seen earlier, invariant language allows us to transform bisequents into four-valued formulas, which means that four-valued logics in languages that include the invariant connectives can be given a standard Hilbert-type axiomatization. Moreover, since the locally classical connectives behave in

106


a fully classical way in this setting, such axiomatizations can be constructed as extensions of classical logic with additional connectives. A four-valued logic U4 in the language with the invariant connectives {A, ->, ~} is determined by the axioms (tautologies) and rules of classical logic for the language {A, ->}, plus the following axioms for ~ (where • is the usual classical equivalence denned in terms of {A, ->}): Double Negation ~~yi A; Commutativity ~-~J4; De Morgan ~(A A 5 ) f > ~ A V ~JB. Though the logic L 4 is an extension of classical logic, it should be noted that the global negation ~ lacks the usual property of replacement of provable equivalents. In particular, 4 e B does not imply ~A ~ B . By u \-\ A we will denote the fact that A is derivable from the set u of propositions in L4. Given the correspondence between bisequents and formulas in the invariant language, established in Sec. 4.1.6, it is straightforward (though tedious) to show that the logic L 4 provides a complete axiomatization of four-valued reasoning in the invariant language: Theorem 4.20

For any set u of propositions in the invariant language, u\-\ A if and only if u 1=4 A.

Moreover, any additional four-valued connective can be incorporated into this logic by adding translations of the corresponding rules for this connective in biconsequence relations. For example, an axiomatization of A can be obtained by adding the following two axioms: AA

• ~-i.A,

~ A A • ~ / L

Actually, adding A to the invariant language results in a functionally complete set of classical four-valued connectives, so any such connective is already definable in this setting. For example, since — A — ~A^4, the ht-negation — is axiomatizable as follows: -A

• ~A,

~ - A A. Finally, it is interesting to note that the logic L 4 is actually equivalent to the Nonstandard Propositional Logic (NPL) from [Fagin et al., 1995].

Four-Valued Logics

107

Using our notation, NPL has been defined in the language {A, ~, ->}, where —>• is a material implication definable in {A, -i}. Note, however, that the local negation ->A is definable in the language of NPL as A —» ~(A —> A), and consequently the language of NPL turns out to coincide with the invariant language. Moreover, under this reformulation, the axiomatization of NPL, given in [Fagin et al., 1995], will almost coincide with the above axiomatization of L4. The equivalence of the two logics will also immediately follow from the semantic interpretation, given below. 4.3.2

Possible worlds

semantics

We describe now an important alternative semantic representation for biconsequence relations and four-valued logics in invariant languages. The origins of this representation are in the so-called 'Australian Plan' semantics for relevant logics (see [Routley et al., 1982]), as opposed to the four-valued 'American Plan' semantics ([Dunn, 1976; Belnap, 1977]). Using the characteristic condition of invariant connectives from Lemma 4.5, we immediately obtain that, for any valuation v and any formula A built with invariant connectives only, we have v =j A iff v* £ A. The above equivalence says that a proposition A is false in a four-valued interpretation v if and only if it is not true in the dual valuation v*. This means, in effect, that, if the notion of a dual valuation is given, the falsity valuation for an invariant proposition can be defined in terms of truth assignments with respect to a given and/or dual valuation. This leads to the following construction: Definition 4.5 A possible worlds *-model is a quadruple W = (W,N,*,V), where W is a set of possible worlds, N C W is the set of normal worlds, V is a valuation function assigning each world from W a (classical) propositional interpretation, while * is a function on W such that, for any a G W, a** = a. In what follows, a \= A will denote the fact that the proposition A holds in the possible world a. By the intended understanding of the above *-model, a* determines a valuation that is dual to a, and therefore pairs (a, a*) stand in one-to-one correspondence with four-valued interpretations. Accordingly, we can give a semantic characterization of the invariant connectives in this setting. In

108


fact, it is sufficient to give corresponding semantic definitions only for the base set of the invariant connectives {A, ->, ~ } . These are as follows:

a\=AAB iff a\= A and a\=B a\=->A iff c * M a\=~A iff a*tf=A. As can be seen, the local connectives coincide with ordinary classical connectives in this setting, while the global negation coincides with the so-called relevant negation (see [Routley et al., 1982]). The intended interpretation of *-models immediately suggests also the following definition of validity for bisequents in such models. Definition 4.6 A bisequent a:b Ih c:d will be said to be valid in a possible worlds *-model W if, for any normal world a £ N, if a \= A, for any A £ a, and a* ^ B, for any B £ b, then either a\=C, for some C £ c, or a* y= D, for some D G d. For a *-model W, we will denote by Ihw the set of bisequents that are valid in W. It could be easily verified that the latter set always forms a biconsequence relation. Moreover, the following theorem establishes the corresponding completeness result. Theorem 4.21 II- is a biconsequence relation in the invariant language if and only if ]\-—]\-^, for some *-model W. Proof. We will establish a direct equivalence between *-models and fourvalued semantics for the invariant language (cf. Sec. 3.2 in [Routley et al., 1982]). For a *-model W = (W, N, *, V), the associated set of four-valued interpretations Iw will be the set of all interpretations i/a, for a £ N, defined as follows: for any propositional atom p,

va\=p = a\=p and va =)p = a* \fcp. As can be verified (by induction on the complexity of formulas), the above equivalences are extendable to arbitrary invariant formulas. Namely, for any invariant formula A we have

va\=A\Sa\=A

and va =| A iff a* £ A.

As a result, we immediately obtain that W and Iw determine the same sets of valid bisequents. In the other direction, for a set of four-valued interpretations I, we construct the associated *-model Wi as follows:

Four- Valued Logics

109

• Every i / g l will determine two worlds v+ and J/_ . W will be denned as the union of all such worlds, while N as the set of all worlds of the form u+. • The function * will be defined in an obvious way: 1/+ — V-

V*_ — V+.

• The valuation function V on W will be defined as follows: v+\=p = v\=p v-\=P = vi\pAgain, it can be easily verified that the *-model Wi determines the same set of valid bisequents in the invariant language as I. This shows that the two semantics are fully equivalent for biconsequence relations in the invariant language. Due to the equivalence between the two semantics, we obtain that the possible worlds *-semantics is also adequate for biconsequence relations in the invariant language. • As a consequence of the above result, we obtain also that the ^-semantics provides an adequate interpretation of the four-valued logic L 4 . Definition 4.7 A proposition A will be said to be valid in a *-model W if a \= A, for any normal world a. Then we have Corollary 4.22 For any set u of propositions, u \-\ A if and only if A is valid in any *-model that validates u. To conclude this section, we will provide a direct description of the canonical possible worlds models for L 4 . Such canonical models will be used in subsequent chapters. For a given 4-world a, we will define the following set of propositions: a* = {A | ~ A £ a } . It can be verified that if a is determined by a four-valued interpretation 1/, then a* will be a 4-world determined by the dual interpretation v*. Let TV be a set of 4-worlds. We will denote by N* the set of dual 4worlds {a* \ a G N}. Then the corresponding canonical possible worlds * -model will be denned as a *-model W = (N U N*, N, *, V), where the function * is defined as above, while V is defined as follows: a 1= p iff p £ a. Then, for any formula A, a 1= A will hold if and only if A 6 a.

110

4.3.3


Invariant

inference

Recall that the invariance principle requires that four-valued reasoning with respect to truth should coincide with reasoning with respect to non-falsity. A formal expression of this principle has been given in the preceding chapter using the Invariance rule. In this section we will give a more detailed description for such an invariant four-valued reasoning. To begin with, we will show that invariant four-valued reasoning amounts to preservation of a truth order among the truth-values: f A Ih. But the latter is reducible to Ih: A. Thus, Positive Coherence implies • Negative Coherence. The reverse implication is proved similarly. L e m m a 4.28 //Ih is a coherent biconsequence relation in a language C with the global negation ~ , then, for any proposition A in C, • Ih A : if and only if: A Ih; • A : I h if and only if I h : A. Proof. If : A Ih, then ~A : Ih in the language extended with connectives from £, and hence Ih: ~A by Negative Coherence. The latter bisequent is reducible to Ih A :. Similarly, it can be shown that Ih: A implies A : Ih. • Thus, for biconsequence relations that are coherent in languages containing ~, provable truth coincides with provable classical truth and provable non-truth (refutability) coincides with provable falsity. In what follows, we will give a structural description of coherent biconsequence relations in different languages. As we will see, this description will cover practically all kinds of biconsequence relations described earlier. {V, A}-coherence If the language C contains no connectives, the coherence rules coincide with their structural counterparts. If the language contains disjunction,

116


Positive Coherence is already equivalent to a 'multiple' structural rule given below, though Negative Coherence is still reducible to its singular variant. Adding conjunction will give a corresponding multiple variant of Negative Coherence: L e m m a 4.29 A biconsequence relation Ih is {V, A\-coherent iff it satisfies the following structural rules: II-a: :al(-

a:lh Ih : a

Proof. Since any finite set of propositions is replaceable by its conjunction in positive premises and negative conclusions, and by its disjunction in negative premises and positive conclusions, the implication from left to right is obvious. To prove the reverse implication, we will show a stronger result, namely that if the above structural rules hold for a biconsequence relation, then the corresponding logical rules in the language {V, A} will also hold with respect to it. This can be proved by induction on the total number of conjunctions and disjunctions occurring in these rules. If Ih a,A/\B :, then IH a, A : and Ih a, B :, and therefore by the inductive Ih by the assumption : a, A Ih and : a,B Ih. Consequently, : a,A/\B properties of conjunction. If Ih a, A V B :, then Ih a,A,B: by the properties of disjunction. Hence : a,A,B Ih by the inductive assumption (note that a U {A} U {B} contains less connectives than a U {A V B}). But then : a, A V B Ih. In the same way it can be proved that the second structural rule implies its {V, A}-logical counterpart. Now, the relevant {V, A}-coherence rules are simply special cases of such logical rules, and hence the implication from right to left also holds. • {V, A,

L}-coherence

The following result describes structural equivalents for {V, A, Incoherence rules. Lemma 4.30 {V, /\,\j)-coherence the following structural rules: Ih a, b : :alh&:

rules are equivalent, respectively, to a, b : Ih a : Ih : b

Proof. We will consider only Positive Coherence here; the proof for Negative Coherence is completely analogous.

Four-Valued Logics

117

If Ih a,b : then Ih \J(a U U) : by the properties of disjunction and L. Hence : \/(aULb) Ih by Positive Coherence, which is equivalent to : a Ih b :. Thus, Positive Coherence implies the corresponding structural rule. To show the reverse inclusion, we will prove that this structural rule implies its {V, A,L}-logical counterpart. Again, this can be done by induction on the total number of connectives occurring in propositions of the rule. We will consider only the case of L. If Ih LA, a, b : then Ih A, a, b :. Applying the inductive assumption, we obtain : a Ih A, b :. Therefore, both : LA, a Ih b : and : a Ih b, LA : hold due to the properties of L. This gives us the two cases of the rule depending • on whether LA is adjoined to a or to b.

Local coherence For {A, -i}-coherence, that is, coherence with respect to the local connectives, Positive Coherence and Negative Coherence are already equivalent. Moreover, we have Lemma 4.31

Local coherence is equivalent to the structural rule a :Ih c : :clh:a

Proof. If a :lh c :, then II—>a,c :, and hence Ih V("nQ u c ) - Applying Positive Coherence, we obtain : V(->aUc) Ih, which is reducible to : c Ih: a. Thus, Positive Coherence implies the above structural rule. In the other direction, it can be proved that this structural rule implies the corresponding {A, -i}-logical rule (again, by induction on the complexity of propositions occurring in it). Since both Positive and Negative Coherence are special cases of such a logical rule, this will complete the proof. • The above structural rule corresponds to an interesting semantic constraint on possible interpretations. It says that, for any bitheory (u, v) there is a bitheory of the form (v,w). In other words, any negative part of an admissible interpretation should also serve as a positive part of some other interpretation. A strengthening of this constraint to the requirement that if (u, v) is a bitheory, then (v, u) is also a bitheory will give us a semantic description of invariant biconsequence relations (see below).

118


Regular coherence Regularity turns out to be equivalent to coherence in the intermediate language {A,->,-}. Lemma 4.32 A biconsequence relation is regular if and only if it is {A, ->, —^-coherent. Proof. If a:b Ih c:d, then Ih V(~' a U -• - 6 U c U - d ) : (see Lemma 4.10 (5)), and hence : V ( ^ a U -i — 6 U c U — d Ih by coherence. By the properties of -i and —, the latter bisequent is reducible to : b, c Ih: a,d, and therefore Regularity holds. As before, it can be proved also that the structural Regularity rule implies the corresponding {A, ->, — }-logical rule (by induction on the complexity of propositions), whereas the Coherence rules are just • special cases of such a logical rule.

Conservative

coherence

The following result shows that Consistency and Completeness are also kinds of coherence rules. More exactly, they are equivalent to coherence rules in the conservative language. Lemma 4.33 A biconsequence relation is consistent (respectively, complete) if and only if it satisfies Positive (resp. Negative) Coherence with respect to the conservative language. Proof. Since Ih ~LA V A : is a valid bisequent, we have : ~ L A V A Ih by Positive Coherence. The latter bisequent is reducible to A : A Ih, and therefore Positive Coherence in our case implies Consistency as a logical rule in the conservative language. In the other direction, it is easy to show that consistent biconsequence relations make valid bisequents A : A Ih for all conservative propositions A (by induction on the complexity of A). Consequently, if Ih A : holds, we obtain : A Ih by Positive Cut. Thus, Positive Coherence holds. • The proof for completeness is similar.

Invariant coherence Now we will show that Invariance is equivalent to coherence with respect to the language of invariant connectives. Lemma 4.34 A biconsequence relation is invariant if and only if it is coherent in the invariant language.

Four-Valued Logics

119

Proof. If a : b Ih c : d, then Ih \/(-,a U ->~6 U c U ~d) : (see Lemma 4.10 (2)). Applying coherence, we obtain : \J(-b U ->c) | a : b Ih c, A : d € A}, plus the set {-. /\(a U d U -16) | a : b Ih : d 6 A}. L e m m a 5.58 / / A is a locally finite acyclic bisequent theory, then the classical models of comp(A) correspond exactly to extensions of A. Now, since autoepistemic bisequent theories are trivially acyclic, we immediately obtain Corollary 5.59 The default nonmonotonic semantics of a locally finite autoepistemic bisequent theory coincides with the classical semantics of its completion. Clearly, the completion of an autoepistemic bisequent theory A is determined as the set of all classical formulas of the form A^\J{/\(dU^(bl)c))

\:b\\-c,A:d€A},

plus the set {-. f\(d U -.6) |: b Ih : d £ A}. Due to the fact that expansions of an acyclic bisequent theory coincide with its extensions, the above results give us, in effect, all we need in order to prove the following theorem. Theorem 5.60 If A is a locally finite bisequent theory, then the classical models of comp(A) correspond exactly to expansions of A. Proof. Recall that any bisequent theory A is logically reducible under the supported semantics to the autoepistemic bisequent theory auto(A). Consequently, the plain completion of auto(A) corresponds exactly to the

144


set of expansions of A. Now the result follows from the fact that A and auto(A) determine the same completion. •

5.5

Partial Nonmonotonic Semantics

In this section we are going to describe a generalization of the nonmonotonic semantics, described earlier, to partial models. Partial nonmonotonic semantics have played a prominent role in logic programming, and there have been numerous attempts to extend them to general nonmonotonic formalisms. In this section we will give a structural description of such partial semantics. In Chapter 8 we will provide these semantics with a logical basis. All the nonmonotonic semantics, described earlier, have been defined as certain sets of theories of a biconsequence relation. Now, while a biconsequence relation can be viewed as a representation of a possibly incomplete and/or inconsistent information, its theories correspond to 'classical' models of the latter, since they are precisely bitheories that are both consistent and complete. In other words, while biconsequence relations are inherently partial, its nonmonotonic semantics, as defined above, are essentially classical. This conceptual incoherence naturally suggests that it is worth exploring a possibility of generalizing the nonmonotonic semantics to partial models. In other words, it seems interesting to consider generalizations of these semantics that are based on a broader set of bitheories of a biconsequence relation, instead of just its theories. Three guiding principles will be used below in denning partial counterparts of nonmonotonic semantics. First, it is natural to require that such a semantics should subsume the corresponding 'classical' nonmonotonic semantics. Second, it should be required that partial semantics must preserve the explanatory nature of the original nonmonotonic semantics, namely, the assumption context of each partial model should explain its corresponding objective (positive) part. Third, we will require, in addition, that the intended partial models should be symmetric with respect to its components, namely they should correspond to invariant bitheories of a biconsequence relation. The last requirement will increase the similarity with the earlier nonmonotonic semantics. It will give us also partial models corresponding to existing semantics of logic programs. The following definition describes the partial counterparts of the main nonmonotonic semantics that conform to the above requirements.


Definition 5.14 Ih will be called

145

An invariant bitheory (u, v) of a biconsequence relation

• partial exact, if there are no other sets u\ ± u and v\ / v such that (vi,u) or (ui,v) are bitheories of Ih; • a partial extension, if both (u,v) and (v,u) are positively minimal bitheories of Ih; • a, partial expansion, if (u\{A},v) and (v\{B},u) are not bitheories of Ih, for any A 6 u and B g w. In accordance with our earlier terminology, the set of partial exact bitheories will be called a partial exact semantics of Ih, the set of partial extensions will form a partial default semantics of Ih, while the set of partial expansions will form a partial supported semantics of Ih. As before, all these partial semantics will be extended to arbitrary bisequent theories in an obvious way. Partial nonmonotonic semantics were defined above as certain sets of bitheories. Each such bitheory can be considered as a four-valued interpretation, which gives us yet another dimension of freedom in determining partial nonmonotonic consequences of a bisequent theory. Thus, if («, v) is a bitheory, then propositions from u fl v can be viewed as classically true (namely, both true and non-false), propositions from u fl v can be viewed as classically false (both false and non-true), propositions from u fl v are undetermined (neither true, nor false), while propositions from Mflvcan be seen as contradictory (both true and false). In other words, while the earlier 'classical' nonmonotonic semantics, being sets of theories, determined Scott consequence relations, the informational content of partial nonmonotonic semantics is described by the biconsequence relation generated by the corresponding set of bitheories. This biconsequence relation clearly provides a more fine-grained description of the information that is nonmonotonically implied by a source bisequent theory or biconsequence relation. A more specific feature of the definitions given to the partial semantics is that each of them forms a symmetric binary semantics: L e m m a 5.61 / / (u, v) is a partial exact bitheory (partial extension, partial expansion) of a biconsequence relation, then (y, u) is also a partial exact bitheory (resp., partial extension, partial expansion). As a result, the biconsequence relations generated by the partial nonmonotonic semantics will always be invariant (see Sec. 3.2.7). It is important to note that the information provided by the classical

146


nonmonotonic semantics is not lost in this setting, and it can be restored just by restricting the corresponding set of bitheories to theories, that is, to bitheories that are both consistent and complete. This is an immediate consequence of the following simple result. Lemma 5.62 A theory u of a biconsequence relation is exact (an extension, an expansion) if and only if (w, u) is a partial exact bitheory (respectively, partial extension, partial expansion). The proof of the above result can be discerned directly from the corresponding definitions of nonmonotonic semantics. It shows that restrictions of partial nonmonotonic semantics to classical models gives us precisely the corresponding classical nonmonotonic semantics. Yet another general correspondence between these nonmonotonic semantics and their partial counterparts will be established below using a doubling construction. An additional constraint that is often imposed on partial nonmonotonic semantics consists in choosing only consistent partial models, namely bitheories (u, v) such that u C v. Clearly, such a consistent partial nonmonotonic semantics will still subsume the classical nonmonotonic semantics, so it will occupy an intermediate ground between the classical and fully partial semantics in our sense. Still, such a constraint does not seem to have a systematic justification (apart from the desire to be consistent), so it will not be used in this study. Finally, recall that all the classical nonmonotonic semantics satisfied a restricted monotonicity property according to which addition of bisequents that preserves theories leads to a growth of the corresponding nonmonotonic semantics. A partial counterpart of this property can be obtained, if we restrict possible additions to bisequents that preserve invariant bitheories. Let us say that two bisequent theories F and A are invariantly equivalent, if they determine the same invariant biconsequence relation. In other words, they are inter-derivable using the postulates for biconsequence relations and Invariance. Clearly, two bisequent theories are invariantly equivalent if and only if they have the same invariant bitheories. Now, an addition of new bisequents to a bisequent theory results in removal of some of its bitheories. But if the invariant bitheories are preserved, any exact bitheory, partial extension or partial expansion will remain such in the new bisequent theory. Accordingly, we obtain Lemma 5.63 If F and A are invariantly equivalent bisequent theories such that F C A, then any partial exact bitheory (partial extension, partial


147

expansion) of T is a partial exact bitheory (respectively, partial extension, partial expansion) of A. 5.5.1

Doubling

Now we will describe a general translation that will transform every partial nonmonotonic semantics into the corresponding classical nonmonotonic semantics. The translation will be obtained by doubling the set of propositions of the language, and transforming the bisequents into new bisequents in the extended language. Again, the origins of this construction are in logic programming. By a language £ we will mean in this section simply a set of (atomic) propositions. For any proposition A from £, we will extend the language with a new proposition A'. C! will denote the set of all such new propositions, and for any u C £, we will denote by u1 the set {A' \ A 6 u). Finally, we will transform the bisequent theory A into a new bisequent theory A' in the extended language, obtained by replacing every bisequent a:b Ih c:d from A with the following two bisequents: a':b\Vc':d

a:b'\Vc:d'.

The general effect of this translation amounts to a full separation (or 'split') between the two contexts associated with the bisequents of A. As a result, the new bisequent theory will have especially regular logical properties. As before, IH^/ will denote the least biconsequence relation (in the extended language) that contains A'. The next results give a description of this biconsequence relation. Lemma 5.64 A pair (uUu'-y, vUv'i) is a bitheory of A' if and only if both (ui,v) and (u,vi) are bitheories of A. Proof. (u{J u[,v U v[) is a bitheory of A' if it is closed with respect to the bisequents of A'. By the construction of A', this means that, for any bisequent a:blb c:d from A, if a C u\ and b C F, then either c(l u±, or dDv, and if a C u and b C Fi, then either cf)u, or d D «i. The latter conditions amounts to saying that both («i, v) and (u, vi) are bitheories of A. D As a consequence of the above description, we immediately obtain that the bitheories of A' satisfy the following invariance property:

148


Corollary 5.65 //(wUw'j, vUv[) is a bitheory of IS.', then (uiUu' is also a bitheory of A'.

,viUv')

In addition, the above lemma gives the following characterization of theories of A': Corollary 5.66 A set uU u[ is a theory of A' if and only if the pair (u,ui) is an invariant bitheory of A. The corresponding syntactic description of II~A' is a s follows: L e m m a 5.67

For any sets of propositions a, a,i, b, &i, c, ci, d, d\ from C,

a, 0^:6, &'x lt"A' c i c i : î^'i iff a\:b W&. C\:d or a:b\\\-& c:d\. Proof. Let us denote by IK the set of bisequents in the extended language that are determined by \Y& using the above equivalence. Then it can be easily verified that IK is actually a biconsequence relation that includes A'. Consequently, Ih/yClK. In the other direction, if a,a[:b, b\ Vf&i c, c\:d, d[, then there exists a bitheory (w U u\, v U v[) of A' such that a C u, a; C «i, b C v, 61 C TFj", c C u, c\ C u\, d C v and di C vi. But by the preceding lemma (ui,i>) and (u, v\) are bitheories of A, which implies, respectively, that ai:b\>f& c\\d and a:b\ JÂ c:d\. Consequently, a,a'x:b,b\ Vf' c, c\:d, d[, which gives the direction from right to left. • As a consequence of the above description, we obtain that bisequents of IHA' are invariant with respect to the ' operation: Corollary 5.68

For any sets a, ai, b, bi, c, c\, d, di from C,

a, a^'.b, b^ II~A' C, cîd, d± iff a\, a :b\, b II~A' CI, c '.d\, d . Finally, the following theorem shows that the above translation transforms each partial nonmonotonic semantics into its classical counterpart. Theorem 5.69 A pair (u, v) is a partial exact bitheory (partial extension, partial expansion) of a bisequent theory A if and only if uUv' is an exact theory (resp., extension, expansion) of A'. Proof. We will prove the result only for the (partial) exact semantics, the proof for extensions and expansions being similar. uUv' is an exact theory of A' if and only if, for any set MJ U V[, the pair («i U B ' ^ U U V') is a bitheory of A' iff «i U «! = tiUii'. By Lemma 5.64, this amounts to saying that (ui,v) and (ui,«) are bitheories of A if and only if Mi = u and vi = v. The latter is equivalent to the claim that (w, v) is a partial exact bitheory of A. •

.

149


The above result allows us to systematically translate all the questions and problems about partial semantics into questions about their classical counterparts. In addition, the reader should note similarity between the above doubling construction and the classical translation of four-valued logics, given in Sec. 4.1.7 of Chapter 4. The similarity suggests that partial nonmonotonic semantics can be systematically reduced to the original nonmonotonic semantics in some generalized 'partial' four-valued logical formalism. Such a formalism will be described in Chapter 8 in the framework of causal inference. For the time being, in the following sections we will give a more detailed description of the partial semantics.

5.5.2

Partial exact semantics

As follows from the above definition, a pair (u, v) is a partial exact bitheory of Ih if and only if, for any sets ui, Vi of propositions, («i, v) is a bitheory of Ih if and only if wi = u, and (•ui, u) is a bitheory of Ih if and only if v\ = v. Thus, just as for exact theories, partial exact bitheories are bitheories, for which both u and v, taken as assumption contexts, determine the second set as a unique objective state compatible with it. In other words, the positive u-reduction of Ih, lh+, is a unitary consequence relation having v as its only theory, while u is similarly the only theory of lh+. Consequently, Lemma 2.27 implies the following syntactic description of partial exact bitheories. L e m m a 5.70 A pair (u,v) is a partial exact bitheory of a hiconsequence relation Ih if and only if, for any proposition A, A £ u iff

:v\\- A : v

A £ u iff A : v Ih : v

A £ v iff

:u\VA:u

A £ v iff A : u \ V : u .

The corresponding description of the partial exact semantics for bisequent theories can be obtained using again Lemma 3.9. L e m m a 5.71 (w,u) is a partial exact bitheory of a bisequent theory A iff v is the only theory of A J , and u is the only theory of A+. The next example shows that a bisequent theory can have partial exact models even when it does not have theories.

150


Example 5.3 A bisequent theory :B\VA:A \\-A:A,B :A,B\\-A:

: B Ih B : A B:H-:A,B : A Ih A : B

does not have any theories. In other words, it is thoroughly classically inconsistent. Nevertheless, it has a unique partial exact bitheory ({A,B},{A}). Unfortunately, the above example shows also that Regularity is no longer appropriate for partial exact semantics. Indeed, Regularity requires that (u, v) is a bitheory only if v is a theory of a biconsequence relation. But the above bisequent theory does not have theories, so imposing the Regularity postulate would make it inconsistent. Actually, the failure of Regularity gives a first indication of the fact that a proper description of the underlying logics for partial nonmonotonic semantics cannot be given directly in the framework of biconsequence relations. Nevertheless, the doubling construction will allow us to give a corresponding semantic description. Theorem 5.72 Bisequent theories T and A are strongly equivalent with respect to a partial exact semantics iff (i) V and A are invariantly equivalent, and (ii) if (u, v) is an invariant bitheory ofT, then (w, u) is a bitheory of F iff it is a bitheory of A. Proof. To begin with, due to Lemma 5.64, it can be verified that the above two conditions are satisfied iff F' and A' have the same regular bitheories. Consequently, by Theorems 5.5 and 5.69, if F and A satisfy the above conditions, they determine the same partial exact semantics. Moreover, for any set of bisequents $ we have also that V U f :v, are required,respectively, only when v or u is the set of all propositions. A direct consequence of the above descriptions is that partial extensions are determined by the autoepistemic subrelation of a biconsequence relation. Corollary 5.75 The partial default semantics of a biconsequence relation coincides with the partial default semantics of its autoepistemic subrelation. For singular biconsequence relations, the above description of partial extensions is simplified as follows. The proof follows immediately from Corollary 3.36. Corollary 5.76 (w, v) is a partial extension of a singular biconsequence relation Ih if and only if u = {: v Ih A : v} and v = {: u Ih A : u], and either u, v / Pr, or Vf: Pr. A description of the partial default semantics for a bisequent theory is given in the next lemma. The proof is immediate from the fact that (v, u) is a bitheory of A if and only if v is a theory of A+.


153

Lemma 5.77 (u, v) is a partial extension of a bisequent theory A if and only if u is a minimal theory of A+, and v is a minimal theory of A+. As with the exact semantics, the correspondence between extensions and partial extensions cannot be extended to the corresponding logics. Thus, Consistency is clearly inappropriate for partial extensions; this follows already from the fact that the restriction to consistent bitheories reduces all invariant bitheories to ordinary theories. The failure of Consistency also abolishes a direct correspondence between expansions and exact theories for the partial case. Still, the doubling construction will give us the corresponding semantic description in this case too. Theorem 5.78 Bisequent theories F and A are strongly equivalent with respect to a partial default semantics iff (i) T and A are invariantly equivalent, and (ii) if(u,v) is an invariant bitheory of T, and w C v, then (w,u) is a bitheory of F iff it is a bitheory of A. Proof. Due to Lemma 5.64, the above two conditions are satisfied iff F' and A' have the same consistent regular bitheories. Now, Theorem 5.36, coupled with Theorem 5.69, implies that if the above conditions are satisfied, F and A will be strongly equivalent. Assume now that the above conditions are not satisfied, and suppose first that («, v) is an invariant bitheory of F that is not an invariant bitheory of A. If u — v, the proof reduces to the corresponding part of the proof of Theorem 5.36. So suppose that u ^ v, and assume for certainty that (u, v) is not a bitheory of A and, moreover, v ^ u (the other cases being similar). Then there exists Ao G v\u. For this case we denote by

iff at:b\-^

or a:b\ KA or a:b\ I~A •

Proof. The implications from right to left are immediate. Let h' denote the set of default rules in the extended language that are determined by HA using the above equivalences. Then it is easy to verify that h' is actually a default consequence relation that includes A'. Consequently, h A ' C K , • which gives the direction from left to right. Finally, the following theorem shows that the above translation transforms each partial nonmonotonic semantics into its original counterpart. Theorem 6.20 (u, v) is a partial extension (partial expansion) of a default theory A if and only if u U v' is an extension (resp., expansion) of A'. Proof. We will prove the result only for (partial) expansions, the proof for extensions being similar. iiUn' is an expansion of A' if u U v' = Q I A ' ( « U v':u Uti'). The latter equality amounts to the following two conditions: A € u iff M U B ' : ullv' A€v

r-A< A

iff u U t i ' : « U » ' r-A' A'.

By the preceding lemma, these conditions amount, in turn, to A £ u iff u : v I-A A or v : u HA A G v iff v : u HA A or u : ¥ HA .

184


Note, however, that, since 11U0' is an expansion, it is a theory of A', and consequently both (u, v) and (v, u) should be bitheories of A by Corollary 6.18. Consequently, the above conditions are reducible to i£«

iff u : v

A E. v iff v : u

I~A HA

A. A.

which say that (w, v) is a partial expansion of A.

•

As we already mentioned in the preceding chapter, a logical basis of the above doubling construction will be provided in Chapter 8 in the framework of causal inference.

6.4

Kinds of Default Reasoning

The general notion of a default consequence relation is rather uninformative; it is only a frame that can be filled with additional rules that would provide a more tight description of our intuitions about nonmonotonic reasoning. As we will see, however, there will be no single system that reflects adequately all our intuitions, since different nonmonotonic semantics will admit different underlying logics. We will present now three rules for default consequence relations that will form the basis for a subsequent classification of different kinds of default reasoning.

Reflexivity A : h A. Consistency A: A\~. Factoring

If a, B : b h A and a : b, B h A, then a :b\~ A.

A default consequence relation will be called reflexive, if it satisfies Reflexivity. Such consequence relation will turn out to be appropriate for nonmonotonic reasoning based on extensions. Reflexivity will be inappropriate, however, for reasoning about expansions. A default consequence relation will be called consistent if it satisfies Consistency. The latter postulate states that no proposition can serve as both a positive and negative premise in a derivation. Finally, a default consequence relation will be called complete if it satisfies Factoring. This postulate permits reasoning by cases with respect to positive and negative premises, and hence presumes completeness with respect to corresponding contexts. This feature turns out to be characteristic


185

of autoepistemic reasoning. The next definition will provide us with semantic counterparts of the above rules. Definition 6.9

A default semantics S will be called

• reflexive, if for any model (w,u,v) from S, u = w; • consistent, if for any model (w, u, v) from S, u C v; • complete, if for any model (w, u, v) from §, v C u. The following result is a combined completeness theorem for the above three kinds of default consequence relations. Theorem 6.21 A default consequence relation is reflexive (consistent,complete) if and only if it has a reflexive (consistent, complete) default semantics. Proof. As in the proof of the general completeness theorem, the direction from right to left amounts to a straightforward verification that the set of default rules determined by a reflexive (consistent, complete) default semantics satisfies Reflexivity (respectively, Consistency and Factoring). In order to provide a uniform and modular proof for the other direction, we have to restrict the canonical semantics, described in the earlier completeness proof. A bitheory (u, v) of h will be called canonical, if (u, v) is a maximal pair such that u:v V A, for some A. As shows the proof of Lemma 6.1, canonical bitheories are still sufficient for determining the source default consequence relation, and hence they also can serve as a basis for a canonical default semantics. Now, it is easy to show that if a default consequence relation 1- is reflexive or consistent, this canonical default semantics is also, respectively, reflexive or consistent. For complete consequence relations, we have to show that if (u, v) is a canonical bitheory, then v C u. Indeed, suppose that there exists some B £ v\u. Then we will have u,B:v t- A and u:B,v h A (since (u,v) is canonical), which implies u:v h- A by Factoring, contrary to our assumptions. Thus the canonical default semantics for h is complete. D It is important to note that the proof of the above theorem shows also that any combination of the three basic postulates is characterized semantically by the corresponding combination of semantic conditions. In the following sections we will consider in more details some important combinations of these postulates.

186

6.5


Default Scott Consequence Relations

We begin with default consequence relations that satisfy all the above rules. Definition 6.10 A default consequence relation will be called a Scott one if it satisfies Reflexivity, Consistency and Factoring. As follows from the semantic characterization of the corresponding rules, the default semantics of a default Scott consequence relation should have only models of the form (u, u,u). Moreover, the following theorem shows that such default consequence relations are actually reducible to ordinary Scott consequence relations. Theorem 6.22 h is a default Scott consequence relation if and only if there exists a plain Scott consequence relation ho such that a :b\- A iff a\-0 b,A. Proof. It is easy to check that if ho is a Scott consequence relation, then the above description determines a default Scott consequence relation. In the other direction, if h is a default Scott consequence relation, let ho be a consequence relation defined as follows: a ho b = a : b h . Due to Consistency and Factoring, this is a Scott consequence relation. We will show that the condition of the theorem is satisfied for hoAssume first that a:b h A. We have A:A h by Consistency, and hence a:b, A h by Cut. Assume now that a:b, A h. Then a:b, A h A by Constraint. In addition, A, a:b h A by Reflexivity, and therefore a:b h A by Factoring. Hence the condition of the theorem is satisfied. • The importance of default Scott consequence relations for our study stems from the fact that the rules Reflexivity, Consistency and Factoring preserve theories of default consequence relations. Let us denote by hs a least default Scott consequence relation containing a given default consequence relation h. Lemma 6.23 h s is a greatest default consequence relation having the same theories as h. Proof. Let T be the set of all theories of h, \-f a Scott consequence relation determined by T, and h* the default Scott consequence relation

187


corresponding to \~r by Theorem 6.22. As follows from this construction, h* could be defined directly as follows: a:b\-*

A

a:bh*

iff

(Vu€T)(aCu&

bCuÂû)

iff ->(3u G T){a C u & 6 C u).

It can be easily verified that h is included in h*, and they have the same theories. We will show that (-* coincides with \-s. The inclusion (-'Ch* follows from the fact that h is included in h*. Assume now that a : b Ks A, and let (w, v) be a maximal bitheory of M such that a C u, b C. v and w : U Y-" A. As in the proof of the completeness theorem, Factoring and Consistency imply that u = v, and hence u is a theory of H. Moreover, by Reflexivity, A (£ u. Now, since hChsCh*, we have that hs has the same theories as K Hence u is also a theory of h, and consequently a : b Y-* A. Thus, P C P , and hence K = P . • An immediate consequence of the above result is the following Corollary 6.24 Default consequence relations h and \~i have the same theories if and only if\-st^=\-s. Now we will turn to the nonmonotonic semantics of default Scott consequence relations, which will be quite simple. To begin with, due to Reflexivity, any theory of such a consequence relation will already be an expansion (since we will always have u C Cn(w:U)). In addition, extensions in this case will be just minimal theories. Lemma 6.25 A set u is an extension of a default Scott consequence relation \- if and only if u is a minimal theory o/h. Proof. Given the representation of Theorem 6.22, the result follows immediately from Lemma 2.18. D The above result shows that default Scott consequence relations (just as ordinary Scott consequence relations) is an appropriate framework only for a simplest kind of nonmonotonic reasoning, namely reasoning with respect to minimal models. It is interesting to note, however, that applications of default-type systems to solving traditional mathematical problems (e.g., the marriage problem—see [Marek et al., 1990]) give rise to just this kind of nonmonotonic reasoning. Though default Scott consequence relations are inappropriate for nonmonotonic reasoning in its full generality, they constitute an important

188


upper bound on reasonable default-type systems. In other words, for reasons that will become clear in what follows, all such systems should contain only rules that are valid for default Scott consequence relations. Accordingly, the simplest way to obtain a nontrivial default consequence relation consists in rejection of one of the three rules constituting the definition of a default Scott consequence relation. In this respect, extensions and expansions give rise to two different kinds of reasoning. An extension-based reasoning is characterized by rejecting Factoring and accepting the other two rules. Such a reasoning is associated with default logic and modal nonmonotonic logics based on 'negative introspection'. An expansion-based reasoning amounts to rejection of Reflexivity, and is associated with autoepistemic logic. Below we will give a description of the corresponding systems. 6.6

Reflexive Default Consequence Relations

In this section we will single out the range of default consequence relations that are appropriate for reasoning with extensions. Recall that a default consequence relation is reflexive if it satisfies Reflexivity. A convenient feature of such consequence relations is that the corresponding provability operator Cn(u:v) has all the usual properties of a deductive closure operator with respect to its first argument (that is, when v is fixed): Cl. If u C «!, then Cn(« : v) C Cn(«i : v). C2. U C C H ( U : D ) .

C3. Cn(Cn(w : v) : v) C Cn(w : v). The semantic condition corresponding to Reflexivity is that, in any model (w, u, v) of a default semantics, w must coincide with u. This immediately suggests an alternative semantics for reflexive default consequence relations, namely the binary semantics used earlier for interpreting biconsequence relations (see Sec. 3.1.1). Definition 6.11 A default rule a : b\~ A will be said to be D-valid with respect to a binary semantics B if and only if, for any bimodel (u, v) from B, if a C u and b C v, then A £ u. For a binary semantics B, we will denote by h§ the set of all default rules that are D-valid with respect to B. It is easy to check that this


189

set forms a reflexive default consequence relation. Moreover, the following result shows that reflexive consequence relations are characterized by this semantics. The proof is immediate from the general completeness result, Theorem 6.21. Theorem 6.26 h is a reflexive default consequence relation if and only if there is a binary semantics B such that \~ coincides with I g . As a first consequence of the above result, we conclude that reflexive default consequence relations are uniquely determined by their bitheories. Corollary 6.27 / / r- is a reflexive default consequence relation, then a : b h A if and only if A £ u, for any bitheory (u, v) of h such that a C u and b C v, and a:b h if and only if there is no bitheory (u, v) of \- such that a C u and b C U. Yet another important consequence of the preceding theorem is that default rules a:b h A of a reflexive default consequence relation can be viewed as singular default bisequents a:b IH A :. Formally, we have Corollary 6.28 A default consequence relation is reflexive if and only if it coincides with the set of singular default bisequents of some biconsequence relation. Proof. As can be verified, the set of singular default bisequents of a biconsequence relation satisfies the postulates of a reflexive default consequence relation. Moreover, if h is a reflexive default consequence relation, it has some binary semantics B. Clearly, h coincides with the set of singular default bisequents of the biconsequence relation generated by B. • It should be noted, however, that reflexive default consequence relations trivialize (partial) expansions. Namely, due to Reflexivity, we always have u C Cn(u:v), which immediately implies Lemma 6.29 Any invariant bitheory of a reflexive default consequence relation is a partial expansion. In particular, we have that any theory of a reflexive default consequence relation is an expansion. Consequently, reflexive default consequence relations are already inappropriate for expansion-based nonmonotonic reasoning. Nevertheless, the following result shows that Reflexivity preserves partial extensions, so reflexive default consequence relations will be adequate for the latter.

190


Theorem 6.30 / / P is a least reflexive default consequence relation that contains a default consequence relation h, then P has the same partial extensions as K Proof.

Let us define the following default consequence relation: a : b\~* A = a : b h A or A G a.

It can be directly verified that h* is a reflexive default consequence relation (Cut is the only nontrivial case). Moreover, h* includes h, and hence P C h * . But K is clearly included in P , so P coincides with h*. As a consequence of the above description of P , we have that, for any sets u, v of propositions, Cn*(w:v) = « U &(«:«). In particular, we have Cnt(0:i)) = Cn(0:v). In view of Lemma 6.13, this is sufficient for showing • that P has the same partial extensions as h. Now the correspondence between reflexive default consequence relations and biconsequence relations implies, in turn, the correspondence between (partial) extensions of default theories and (partial) extensions of corresponding bisequent theories. For a default theory A, we will denote by An- the corresponding default bisequent theory, that is, A\\. = {a:b Ih A: | a : b h A € A}. Then we have L e m m a 6.31 For any default theory A, the bisequent theory An- has the same extensions and partial extensions as A. Proof. By the preceding result, partial extensions of A coincide with partial extensions of P A . In view of Corollary 6.28, P A coincides with the singular default subrelation of IÂH-- Note now that, since the latter is a default biconsequence relation, Corollary 3.38 implies that :u lhA|h A:u holds iff :u IFAIH A:. Consequently, (partial) extensions of \\~An- coincide with (partial) extensions of P A due to Corollary 5.76. • The above correspondence gives us, in particular, yet another explanation why extensions are minimal theories of a default consequence relation (see Corollary 5.40).

6.6.1

Regular default consequence relations

We will describe now default consequence relations that constitute a maximal default logic suitable for nonmonotonic reasoning based on extensions. These consequence relations will also play an important role in establishing the correspondence between default logic and modal nonmonotonic logics.


191

It has been established in the preceding chapter that consistent regular biconsequence relations constitute a maximal logic adequate for the default nonmonotonic semantics. In order to transfer this result to default consequence relations, we have to find a default counterpart of Regularity. Definition 6.12 A default consequence relation will be called regular if it is reflexive, consistent, and satisfies the following rule: D-Regulanty

c:6h

{a:b,CÂ\Cec}

• : a : b \- A As follows from the semantic description below, D-Regularity is actually a default counterpart of the Regularity rule for biconsequence relations. In our present setting, however, it is a weakening of the rule Factoring. Accordingly, regular default consequence relations can be seen as a minimal weakening of default Scott consequence relations that is suitable for extension-based reasoning. A binary semantics will be called regular, if it is consistent and quasireflexive (see Sec. 3.2.5). The following theorem establishes a completeness of regular default consequence relations with respect to this semantics. Theorem 6.32 A default consequence relation is regular if and only if it has a regular binary semantics. Proof. It can be easily verified that D-Regularity holds for a regular semantics. Consequently, any regular binary semantics produces a regular default consequence relation. In the other direction, by the general completeness theorem, it is sufficient to show that if («, v) is a canonical bitheory with respect to some A, then v is a theory of K Assume first that v.v K Then c:v h, for some finite c C v. Since (u, v) is canonical with respect to A, we have also uvu, C h i , for any C £ c. Hence by compactness, there are finite a C u and b C ¥ such that c:b rand a:b, C t- A, for any C £ c. Consequently, a:b h A by D-Regularity, and therefore uvv h A, contrary to our assumptions. Hence, vvoY-. Suppose now that v.v V- B, for some B £ v. Then B:U h by Consistency, and therefore v:J> h by Cut, which is impossible. Consequently, Cn(v.v) C v. Thus, v is a theory of h, and therefore the canonical semantics of h is regular. • The next result shows that the rules of a regular default consequence relation preserve extensions. Theorem 6.33 / / h r is a least regular default consequence relation that contains a default consequence relation h, then \~r has the same extensions

192


as h. Proof. We define first a binary semantics B as the set of all bitheories of h of the form (u, u) and (Cn(0:w), u), where u is a theory of K As can be verified, B is a regular semantics, and consequently it generates a regular default consequence relation \~B- Clearly, I- is included in l-g, and therefore KChg, since K is a least such consequence relation. Moreover, any theory of h remains a theory of \-B. Consequently any extension of h will be an extension of he and K by Lemma 6.10. Let u be an extension of hg. Since u is a theory of hg, it is a theory of h, and hence (Cn(0:u), u) £ B. Assume now that A f t i . Then 0 : u \-Q A and consequently A 6 Cn(0:u) by the definition of hg. Therefore, u C Cn(0:w), which means that u is an extension of h. Thus, \~B has the same extensions as h, which immediately implies that both have the same extensions as h r (by Lemma 6.10). • Thus, Reflexivity, Consistency and D-Regularity are rules for default consequence relations that preserve extensions. Now we will show that regular default consequence relations constitute a maximal logic for the extension-based nonmonotonic reasoning. Definition 6.13 Default theories T and A will be called extensionequivalent, if they have the same extensions, and strongly extensionequivalent if, for any set $ of default rules, F U $ is extension-equivalent to AU$. The following lemma is just a reformulation of our previous result in the new terminology. Lemma 6.34 to\-r.

Any default consequence relation h is extension-equivalent

Strong extension-equivalence implicitly determines the logic of default rules that is adequate for the extension-based nonmonotonic semantics. Let us say that default theories T and A are regularly equivalent, if they determine the same regular default consequence relation. This means that each of these default theories is obtainable from the other using the postulates of regular default consequence relations. Then we have Theorem 6.35 Default theories are strongly extension-equivalent if and only if they are regularly equivalent. Proof.

The following proof is similar to the proof of Theorem 5.36.


193

The implication from right to left follows from the preceding lemma. Assume now that F and A are not regularly equivalent. Since reflexive default consequence relations are uniquely determined by their bitheories, we will assume for certainty that (w, v) is a bitheory of h r that is not a bitheory of b A . Since our default consequence relations are regular, uCv, and v is a theory of h r . Suppose first that v is not a theory of h A , and let $ = {(- A | A G v}. Then v will obviously be an extension of H r u $ , though it will not be even a theory of h A u $ . Suppose now that v is also a theory of \-rA. Since (u, v) is not a bitheory of h A , there exists AQ &) | a : 6 h £ A}. Then we have Theorem 6.45 If A a locally finite default theory, then the classical models of comp(A) correspond exactly to expansions of A. The proof follows directly from Corollary 5.59. 6.8

Default Consequence Relations and Normal Programs

Finally, we will show that default consequence relations can serve as a logical framework for normal logic programs. The framework will allow us to give a relatively simple representation of various semantics for such programs, as well as to single out different kinds of logical reasoning that are appropriate with respect to these semantics. Recall that a normal logic program II is a set of program rules A - on finite arguments satisfying the following postulate:

Argumentation Theory

Monotonicity

203

li a'—tb, then a U aj 6 between finite arguments. An argument theory can be viewed simply as an arbitrary relation on finite arguments. Clearly, any argument theory A generates a unique least attack relation that we will denote by A- The latter is obtained from A by closing it with respect to the Monotonicity rule. Accordingly, °->-A can be described directly as follows: u "—»A v iff ao • bo G A, for some ao Cu,boC

v.

An argument theory will be called definite, if it consists of attack rules of the form » 4 A , where A is a single assumption, and singular, if it has only attacks of the form a • b, where b contains no more than one assumption. Then the preceding lemma, coupled with the above representation, immediately implies the following simple observation: Lemma 7.2 An attack relation is local (respectively, normal) if and only if it is generated by a singular (resp., definite) argument theory. Thus, the differences between general, local and normal argumentation are reducible to the differences between corresponding generating argument theories. 7.1.2

Four-valued semantics

Collective argumentation can be given a four-valued semantics that will be instructive in describing the meaning of the attack relation, as well as for imposing plausible constraints on argument theories and their nonmonotonic semantics. The four-valued semantics stems from the following natural understanding of an attack a , we will say that

• a classically attacks b (notation a °->° b) if a, 6a, b; • a negatively attacks b (notation a°->~ 6) if W-â,b; • a positively attacks b (notation a1^"1" b) if a, bu, v iff u, wc->0. This shows that a classical attack amounts to inconsistency in a fully classical sense. It should be noted in this respect that the classical closure ^ ° of an attack relation c—>• preserves consistent arguments of the latter. Namely, the definition of ^-»° immediately implies that it has the same consistent arguments as c—K As could be anticipated, classical argumentation can be characterized semantically by restricting the set of four-valued interpretations to classical two-valued ones, namely to interpretations that assign only t and f to the assumptions. This means that any assumption is either accepted or rejected in an interpretation, but not both. Theorem 7.8 An attack relation is classical if and only if it is determined by a set of classical interpretations.


211

Proof. As can be verified, any set of classical interpretations determines a classical attack relation. Now, if is a classical attack relation, let Ic denote the set of (classical) interpretations corresponding to its bitheories of the form (u, u). Ifw-frv, then u,vu,v, and therefore the interpretation v corresponding to a bitheory (uUv,u\Jv) belongs to Ic. But u (since W-fru,v). Moreover, u'—tv does not hold in the (consistent) interpretation corresponding to this bitheory. As follows from the proof of the main representation theorem, this means that any negative attack relation is determined by a set of consistent interpre• tations. Thus, negative argumentation describes argumentation situations in which assumptions can be accepted, rejected, or neither accepted, nor rejected. A direct expression of this restriction can be given for N-attack relations. Namely, for such attack relations the rule Import is equivalent to the condition


7.3.3

Positive

213

argumentation

The definition below provides a structural description of positive argumentation. Definition 7.9 implies w-^b.

An attack relation will be called positive if a, b-+ will be positive. Moreover, the latter determines a least positive extension of the source attack relation. L e m m a 7.13

+ is a least positive argument theory containing .

The proof is quite similar to the case of negative argumentation, so we will omit it. The lemma implies that positive attack relations are precisely relations of the form ^->+, and hence they give a canonical description of argumentation based on positive attacks. Similarly to negative argumentation, positive argumentation can be characterized by the 'exportation' property described in the lemma below: Lemma 7.14 (Export)

An attack relation is positive iff it satisfies: Ifa,b-+ is clearly both positive and affirmative. Assume that a•—>-+ b,c, that is, a,b,c b.

An attack relation will be called consistent if &•&

As can be seen, consistent attack relations embody the most significant feature of positive argumentation, namely that inconsistent arguments are attacked by any argument. Still, consistency in the above sense is a weaker property than positivity (Export). Clearly, any attack relation of the form •.

Proof. Clearly, ^4-c includes i is a consistent attack relation including t->, and a -i b. In the second case we have b i b, and therefore a i is consistent. Thus, -i should include °->-c, and • consequently it is a least consistent attack relation containing is a normal attack relation, then an assumption A is acceptable for an argument u if and only if A £ [[u]]. Proof. A £ [[«]] holds iff [ u j ^ A The latter is equivalent to the claim that a ^ [u], for any argument a that attacks A. Now, a ^ [u] says that w—}B, for some B 6 a. In the normal case, this is equivalent to if—} a. Hence the result. •

218


Thus, the set of assumptions that are acceptable for u coincides with [[«]]. Note also that, since [ ] is antimonotonic, [[ ]] is a monotonic operator on arguments. Using the above notions, we can give a rather simple characterization of the basic nonmonotonic models of a normal argumentation. Definition 7.16 called • • • • •

For a given attack relation "—>•, an argument u will be

conflict-free if u C [u]; D-admissible if it is conflict-free and u C [[«]]; a complete extension if it is conflict-free and u = [[«]]; a preferred extension if it is a maximal complete extension; a stable extension if u = [u].

An argument is conflict-free if it does not attack any assumption from itself. A conflict-free set u is D-admissible iff any assumption from u is also acceptable for u, and it is a complete extension if it coincides with the set of assumptions that are acceptable with respect to it. Finally, a stable extension is a conflict-free argument that attacks any argument outside it. Clearly, any stable extension is also a preferred extension, any preferred extension is a complete extension, and any complete extension is D-admissible. Moreover, since the operator [[ ]] is monotonic, any Dadmissible argument is included in some complete extension (which is a fixed point of [[ ]]). Consequently, preferred extensions coincide with maximal D-admissible sets. In addition, the set of complete extensions forms a complete lower semi-lattice: for any set of complete extensions, there exists a unique greatest complete extension that is included in all of them. In particular, there always exists a least complete extension of a normal attack relation (or a definite argument theory). As has been shown in [Dung, 1995a], under a suitable translation, the above models correspond to well-known semantics for normal logic programs (see below).

7.4.2

Stable and partial stable semantics

To begin with, note that the above notions and models of Dung's argumentation theory have been defined for arbitrary collective attack relations. It should be clear, however, that they are adequate only for normal argumentation, since they are based only on definite attacks. Still, we will see


219

that in some important cases the more general models defined below will coincide with their normal counterparts. Unfortunately, it will turn out that only a small part of the above nice and well-organized structure of nonmonotonic models for normal argumentation can be transferred into a general framework of collective argumentation. If (u,v) is a bitheory of an attack relation (that is, uu,A}.

Proof. Assume first that u is stable. Then w-fru, and consequently if A G M, then W0-M, A. Also, if A £ u, then u c-û,A, since (w, u) is a

220


negatively minimal bitheory. This gives the direction from left to right. In the other direction, suppose first that u u. Then u 0, which means that is a trivial attack relation. Consequently, W-fru. Suppose now that u'-frv, for some v D u. Then there exists A G U\M, and we have u• coincide with maximal admissible arguments.


225

Proof. (1) Assume that u is admissible, u C v and w—tv. Then wv, u, and hence U),u+ v\u, and therefore u,v• M, them u '—¥ 0, which is impossible, since c—>• is affirmative. Consequently, u is a consistent argument. Also, if v ^->+ w, then u , » H « , which implies u,v-^v\u, and hence u,v'-}v. Thus, w°->-+ v, and therefore u is positively admissible. •

226


Note also that normal logic programs determine normal attack relations. Consequently, a study of weakly stable sets reduces to a study of admissible arguments for what we have called n-positive attack relations. Actually, a large part of Dung's argumentation theory can be reproduced in this setting (see [Bochman, 2003] for further details). It turns out that many other semantics of argumentation, suggested in the literature, are representable as admissibility semantics for particular kinds of attack relations. As an illustration, we will describe below the admissibility semantics for consistent and weakly positive argumentation. An argument will be called consistently admissible, if it is admissible in c-»-c, and weakly positively admissible, if it is admissible in , if u — {A \ w-frA}. As can be seen, the above notion of a stable extension agrees with our earlier definition, though it is defined now on the set of all propositions (instead of just arguments). In the next chapter, this argumentation semantics will be shown to correspond to the main nonmonotonic semantics of causal inference.

2

An alternative, indirect proof of this correspondence will be given in the next chapter.

Chapter 8

Production and Causal Inference

In the preceding chapters we have given a structural description of explanatory nonmonotonic reasoning, namely a description that essentially does not involve propositional connectives and full-fledged logics. In this chapter we will introduce what could be seen as a primary logical system for such a reasoning. The systems of production and causal inference, described below, can be viewed as a most natural and immediate generalization of classical logic that allows to express nonmonotonic reasoning. Basically, this generalization amounts to dropping the Reflexivity postulate of classical inference. The logical foundations of these inference systems will be built in the framework of a logical system for production rules, called production inference relations, that originates in input/output logics of [Makinson and van der Torre, 2000]. It turns out, however, that the latter logics can be assigned both a standard monotonic semantics (that gives a semantic interpretation to production rules) and a natural nonmonotonic semantics, which will allow us to give a fully logical representation framework for the explanatory nonmonotonic reasoning studied in previous chapters. Moreover, it will be shown that biconsequence relations constitute a structural counterpart of this logical formalism. From the point of view of this study, production and causal reasoning constitute the main logical formalisms for explanatory nonmonotonic reasoning. Unlike the epistemic formalisms of default and modal nonmonotonic logics, described later in this study, they are based on a direct and transparent description of factual and causal (explanatory) information about the world. In a hindsight, it even seems strange that the majority of the initial nonmonotonic formalisms suggested in the literature were of epistemic kind, while the causal reasoning has been introduced into nonmonotonic 231

232


reasoning only recently. Anyway, with the addition of causal logics, the picture of explanatory nonmonotonic reasoning will become complete and well-balanced. Our basic language in this chapter will be a classical propositional language with the usual classical connectives and constants {A, V,-•,—>, t , f } . t= will stand for the classical entailment, while Th will denote the associated classical provability operator. As before, p,g,r,... will denote propositional atoms, and A, B, C,... arbitrary classical propositions. 8.1

Production Inference

Production inference relations, introduced below, are based on conditionals of the form A =>• B that hold among classical propositions. A general informal interpretation of such conditionals will be lA produces, or explains, B\ Accordingly, such conditionals will be called production rules. It is worth mentioning that this terminology reflects our conviction that the production rules and associated inference relations constitute a reasonable logical formalization of the well-known, though notoriously vague, notion of production as it has been used in AI. Moreover, we will introduce later a particular, objective interpretation of production rules, according to which the latter will be considered as truly causal rules giving a formal representation of causal reasoning. Though driven by very different considerations and objectives, production inference relations have originated in input-output logics from [Makinson and van der Torre, 2000]. Many results from the latter study will be used in what follows. Moreover, our description of different kinds of production relations in this chapter will closely follow the classification of input-output logics suggested by Makinson and van der Torre. Definition 8.1 A production inference relation is a binary relation =>• on the set of classical propositions satisfying the following postulates: (Strengthening) If A \= B and B => C, then A =^ C; (Weakening) If A => B and B t= C, then A => C; (And) If A^B and A^C, then A=>BAC; (Truth) t=>t; (Falsity) f=^>f. From a logical point of view, the most significant 'omission' of the above set of postulates is the absence of the reflexivity postulate A => A. As we will


233

see in what follows, it is precisely this feature of production and causal rules that will create a possibility of nonmonotonic reasoning in this framework. Production inference relations almost coincide with general input/output logic, except for the last postulate, Falsity. The latter makes the production relations inconsistency-preserving, which amounts, in effect, to restricting the universe of discourse to classically consistent theories. This will give us more smooth semantic characterizations, as well as some additional important properties of production inference. On the other hand, input/output logics constitute, in turn, a special kind of general expectation relations studied in [Bochman, 2001]; the latter have been required to satisfy only Strengthening and Weakening. As a first step, we will extend production rules to rules having arbitrary sets of propositions as premises using the familiar compactness recipe: for any set u of propositions, we define u => A as follows: u=>A = Aa=>A, for some finite a C u. For a set u of propositions, C(u) will denote the set of propositions 'produced' by u, that is C(u) = {A \u=>A}. As could be expected, the production operator C will play much the same role as the usual derivability operator for consequence relations. In particular, it satisfies the following important monotonicity property: Monotonicity If u C v, then C{u) C C(v). Thus, C is a monotonic operator. Actually, due to compactness, C is not only monotonic, but also a continuous operator. Note also that C(u) is always a deductively closed set: Lemma 8.1

For any u, C(u) = Th(C(w)).

Proof. If u =>• A and A t= B, then f\ a => A, for some a C w, and therefore /\a=$B by Weakening, that is, u=$-B. In addition, if A, B G C(u), then /\a=^ A and /\b=5> B, for some a, b C u, and consequently /\(aU6) =>AA5 by Strengthening and And. The latter implies A A B £ C(u). • Still, C is not inclusive, that is, u C C(u) does not always hold. Also, it is not idempotent, that is, C(C(u)) can be distinct from C(u). The following fact will be used in proving completeness theorems for different kinds of production inference.

234


L e m m a 8.2 If A £ C(u), then there exists a consistent deductively closed set v such that u C v, A £ C(v), and A £ C(v'), for any v' D v. Proof. If U is a family of sets that do not produce A, and U is linearly ordered by inclusion, then a usual compactness argument shows that \\U also does not produce A. Consequently, u is included in some maximal set v that does not produce A. Suppose now that v 1= B, though B (£ v. Then A £ C(v,B) due to maximality of v, and hence V A B => A, where V is a conjunction of some propositions from v. But v implies V A B, and hence VQ 1= V A B, where VQ is again a conjunction of some propositions from v. Then VQ =>• A holds by Strengthening, and therefore A £ C(v)—a contradiction. Consequently, v is a deductively closed set. Moreover, f ^ v, • since f => A by Falsity. Therefore, v is also consistent. The reader should note that the theory v in the formulation of the above lemma need not be a world, that is, a complete deductively closed set.

8.1.1

Semantics

We will describe now a general semantic framework for production relations. Our basic semantic object will be a pair of deductively closed theories that will be called, as before, a bimodel. In accordance with the 'input-output' understanding of productions, a bimodel will represent an initial state (input) and a possible final state (output) of a production derivation based on a given set of production rules. The set of such bimodels will give a semantic description for these production rules. Definition 8.2 A pair of consistent deductively closed sets will be called a classical bimodel. A set of classical bimodels will be called a classical binary semantics. Classical bimodels have been defined above in a syntactic fashion, namely as pairs of theories of the language. This formulation will make subsequent constructions simpler and more transparent. Still, any such bimodel (u, v) can be safely equated with a pair of sets of worlds (or propositional interpretations): (w, v) = ({a | u is valid in a}, {p \ v is valid in /?}). All our subsequent constructions will permit such a purely semantic reformulation.


235

A classical binary semantics can also be viewed as a binary relation on the set of deductive theories. Accordingly, given a set of bimodels (semantics) B, we will write uBv to denote the fact that the a bimodel (u, v) belongs to B. These descriptions will be used interchangeably in what follows. Now we will define the notion of validity of production rules with respect to a classical binary semantics. Definition 8.3 A production rule A=>B will be said to be valid in a classical binary semantics B if, for any bimodel (u, v) from B, A £ v only if B e u. We will denote by =>B the set of all production rules that are valid in a semantics B. It can be easily verified that this set is closed with respect to the rules for production relations, and hence we have Lemma 8.3 lation.

For a classical binary semantics B, =3-B is a production re-

In order to prove completeness, for any production relation =>• we will construct its canonical semantics B^, as the set of all classical bimodels of the form {C(w), w), where w is an arbitrary consistent and deductively closed theory. Then the following result is actually a representation theorem showing that this semantics is strongly complete for the source production relation. Theorem 8.4 //S=> is the canonical semantics for a production relation =!>, then, for any set of propositions u and any A, u^-A

iff A G w, for any bimodel (w,v) £ B^, such that u C v.

Proof. If u =£• A and u C v, for some deductively closed set v, then clearly A G C(v). In the other direction, if u =£> A, then by Lemma 8.2, u is included in some theory v such that v j> A. Clearly, (C(v), v) is a bimodel from B^ such that u C v and A £ C(v). This completes the proof. • As an immediate consequence of the above results, we obtain the basic completeness result: Corollary 8.5 A binary relation => on the set of propositions is a production inference relation if and only if it is determined by a classical binary semantics.

236


8.1.2

Causal theories

In what follows, by a causal theory we will mean an arbitrary set of production rules. Since all the postulates for production relations are Horn ones, for any causal theory A there exists a least production relation that includes A. We will denote it by =Â> while CA will denote the corresponding derivability operator. Clearly, =>A is the set of all production rules that can be derived from A using the postulates for production relations. It is important to note that causal theories are undistinguishable, as such, from conditional theories defined in Sec. 2.5 of Chapter 2; the only difference is that causal theories are governed by only a small subset of the classical inference rules. Still, this correspondence will be extensively exploited in what follows when we consider some stronger kinds of production inference. For any set u of propositions, we will denote by A(u) the set of all propositions that can be directly produced from uby A, that is A(u) = {A | B =>• A E A, for some B G u}. The following explicit description of =>A has been given in [Makinson and van der Torre, 2000]: Proposition 8.6 Proof.

C&{u) - Th(A(Th(w))).

Let us define the following production relation: uÂ

= A(Th(w)) PA.

It can be verified that this relation includes A and satisfies all the properties of a production inference relation, and hence we have CA(«) C Th(A(Th(u))), for any u, since =>A is a least such relation. In the other direction, assume A(Th(u)) 1= A. Then A contains a finite set of rules {Bi => Ai} such that y\ A{ P A and up /\Bi. Consequently f\ b P /\ Bi, for some finite b C u. Now we have /\ Bi =>A A ^« by Strengthening and And, which implies /\ b =>A A by Strengthening and Weakening. Consequently, u=>AA, and therefore CA(u) = Th(A(Th(u))). • The above description will be used quite often in what follows. 8.1.3

Regular production inference

A production inference relation will be called regular if it satisfies the following well-known rule:


(Cut)

237

If A => B and A A B => C, then A => C.

Cut is one of the basic rules for ordinary consequence relations. In the context of production inference it plays the same role, namely, allows for a reuse of produced propositions as premises in further productions1. It corresponds to the following characteristic property of the production operator: C(uUC(u)) CC(u). Regular production relations have a number of additional properties. Thus, any such relation will already be transitive, that is, it will satisfy (Transitivity)

If A => B and B => C, then A => C.

Transitivity corresponds to the following familiar property of the production operator: C{C(u)) C C(u). Note, however, that Transitivity is a weaker postulate than Cut, since it does not imply the latter (see below). A production rule of the form A=>f will be called a constraint. Such rules can be used for incorporating a purely factual information into causal theories: a rule A => f says, in a sense, that A is production- or explanatory inconsistent, and hence it should not hold in any intended model. Now, an important property of regular relations is that any production rule implies the corresponding constraint: (Constraint)

If A=^> B, then A A -i£ =^f.

(Indeed, if A^B, then A A -^B^-B by Strengthening and A A B A -•B =S> f by Falsity. Hence A A ->B => f by Cut2.) In particular, we have that t =>• A implies -i^4=>f. Note, however, that the reverse entailment does not hold even in the latter special case. As a special case of Constraint, we have also the rule (Coherence)

If A => -iA, then A => f.

that says that if a proposition produces propositions that are incompatible with it, then it is explanatory inconsistent. Actually, Coherence turns out 1 Such production relations correspond to input-output logics with reusable output in [Makinson and van der Torre, 2000]. 2 Note that the use of Falsity is essential here.

238


to be equivalent to Constraint for all production inference relations (see Lemma 8.50 below). Remark 8.1 Regular production inference sanctions, in effect, an atemporal understanding of the notion of production. For example, the rule p A q => -AA theory of a production relation is a set of propositions that is closed both deductively and with respect to its production rules, namely, if A G u and A => JB, then B £ u. Accordingly, such theories have much the same properties as ordinary theories of consequence relations. Note, in particular, that the set of theories of a production inference relation is closed with respect to arbitrary intersections, and consequently any set of propositions is included in some least such theory. In addition, theories that are worlds have a very simple characterization: L e m m a 8.7 A world a is a theory of a regular production relation if and only if a3>f. Proof. Clearly, if a=>f, then C(a) is an inconsistent theory, and hence C(a) C a cannot hold. Assume now that a world a is not a theory. Then A, ->5 £ a, for some propositions A, B such that A => B. But then we have • also A A -IJB =^f by Constraint, and therefore a =>f. As could be expected, theories of a causal theory A are the sets of propositions that are closed with respect to the rules of A. Thus, the following description follows immediately from Proposition 8.6. L e m m a 8.8 A deductively closed set u is a theory of a causal theory A if and only if A(u) C u.


239

Theories of a regular production relation will determine its canonical semantics, described in the proof of the completeness theorem below. Note, however, that the regular production relation will be determined not only by what its theories are, but also by what they produce. The semantic characterization of regular production relations can be obtained by considering only classical bimodels («, v) such that u C v. We will call such bimodels (and the corresponding semantics) consistent ones. Theorem 8.9 => is a regular production relation if and only if it is generated by a consistent classical binary semantics. Proof. It can be verified that the production relation generated by a consistent semantics will satisfy Cut. In the other direction, for a regular production relation =>, we will construct a canonical semantics as the set of bimodels of the form (C(w),w), where w is a theory of =£>. Clearly, if uÂ and u C w, then A (E C(w). In the other direction, if uj> A, then by Lemma 8.2, u is included in some maximal deductively closed set w such that w=t>A. Assume that w^-B, for some B, though B £ w. Due to maximality of w, we have w U {B} =>• A. Since w is deductively closed, there are propositions C,D 6 w such that C => B and D A B => A. But then by Strengthening and Cut we obtain C A D =$> A, contrary to our assumption that w =£> A. Hence w is a theory of =>, and therefore (C(w), w) is a consistent bimodel that invalidates u =^ A. • The above proof contains, in effect, the following characterization of regular production inference relations in terms of their theories: Corollary 8.10 =^ is a regular production inference relation if and only if, for any set v and any A, v=$-A iff A 6 C(u), for any theory u of =$• that contains v. We will denote by = ^ the least regular production relation containing a causal theory A. As a consequence of the above characterization, we obtain a constructive characterization of =>r&, given in [Makinson and van der Torre, 2000]. Proposition 8.11 contains v.

v =>rA A iff A G Th(A(u)), for any theory u of A that

As a last result in this section, we will show that regular production relations allow to define an appropriate notion of equivalence among propositions such that equivalent propositions would be substitutable in any production rule. Namely, let us say that propositions A and B are production

240


equivalent with respect to a production inference relation, if t =^(^4 • B) holds. Then we have Lemma 8.12 Propositions A and B are production-equivalent with respect to a regular production relation =£• if and only if any occurrence of A can be replaced by B in the rules of =>•.

Proof. If A can be replaced by B in any rule of =>, then it can be replaced also in t =>(A*-»A), which holds by Truth. Hence, t =>(A^-B) holds in =>. We will denote by X(A/B) an arbitrary formula obtained from a formula X by replacing some of the occurrences of A in it by B. Clearly, A^rB 1= I f > I ( A / B ) . Assume now that A and B are productionequivalent, and X =>• Y. Then X =^(A(Y^Y(A/B)) by Weakening. Consequently, X=>Y(A/B) by And and Weakening. Thus, B can replace A in the heads of the rules from =>. In addition, we have X{A/B),AY by Strengthening. But we have also X(A/B)=$(A. • Due to the above result, production-equivalence can be used, in particular, to describe definitional extensions of the underlying language with new propositions (cf. [Turner, 1999]). 8.1.4

Production inference vs. consequence relations

A further insight into the properties of regular production inference can be obtained by comparing it with associated consequence relations. It can be easily discerned from the description of supraclassical Tarski consequence relations, given in Sec. 2.5, that the only difference between the latter and regular production inference relations amounts to the Reflexivity postulate. Note also that any causal theory, and hence any production inference relation, can also be considered as an ordinary conditional theory (a set of inference rules), so it determines a corresponding supraclassical consequence relation. The following construction gives a direct description of this consequence relation in terms of the corresponding production relation. Namely, for a (regular) production relation =>, we will define the following consequence relation: AY-^B

= A^{A->B).


241

Theorem 8.13 // =>• is a regular production relation, then h^ is a least supraclassical consequence relation containing =>. Proof. We will show first that h=> is a supraclassical consequence relation, namely it satisfies Dominance, Transitivity and And (see Sec. 2.5). If A t= B, then A—vB is a tautology, and hence A=$>(A->B) holds by Truth and Strengthening. Thus, Dominance is satisfied. Assume now that A=>(A-tB) and B^B^C. Then A A B^B^C by Strengthening, and hence A=>{B-+C) by Cut. Together with A=>(A-$B), this gives A=$(A—>C) by And and Weakening. Thus, Transitivity holds. Finally, And for h ^ follows directly from the And postulate for =>. Consequently, h ^ is a supraclassical consequence relation. Clearly, => is included in h^. Let hi be any supraclassical consequence relation that includes =>•, and A h ^ B. Then A=$A—yB, and hence A \-\ (A-+B). But A hi A, and hence A hi B. Thus, h^ is included in hi, and therefore it is a least consequence relation containing =>•. • Let Cn=^ denote the provability operator corresponding to h^.. Then the above description can be extended to the following equality, for any set u of propositions: Cn^(u) =Th(uUC(u)). The equality shows, in effect, that C(u) captures practically all nontrivial consequences included in Cn^(u), except for u itself. As a first consequence of the above correspondence, we obtain Lemma 8.14

Theories of =$• coincide with the theories o/h^..

Proof, u is a theory of h ^ iff u = Cn=>(u). By the above equality, this holds if and only if u is deductively closed and C(«) C ti. • Since theories of h ^ are exactly sets of propositions of the form Cn^.(u), the above result implies that such sets are precisely theories of =>. As yet another consequence of the above equation, we obtain the following alternative characterization of regular inference: C{u) =C(C n=> (u)). Note that we have also C(u) — Cn=>(C(u)), so the production operator absorbs Cn^ on both sides: Cn=^ o C = C o Cn=> = C. The above description of associated consequence relations can be immediately generalized to arbitrary causal (= conditional) theories. Thus,

242


if CIIA denotes the least supraclassical consequence relation containing a causal theory A, while CA is the production operator of the least regular production relation containing A, then we have CnA(u) = Th(uUC^(u)). Moreover, Proposition 8.11 will immediately imply the following simplified characterization of CrA: Corollary 8.15

CA{u) = Th(A(CnA(u))).

This characterization will also be used in what follows. 8.2

Nonmonotonic Production Semantics

In the preceding sections, we have given a formalization and standard (monotonic) semantics for the logical system of production inference. It turns out, however, that production inference relations determine also a natural nonmonotonic semantics, and provide thereby a logical basis for nonmonotonic reasoning. Namely, the fact that the production operator C is not reflexive creates an important distinction among theories of a production relation. Definition 8.5

For a production inference relation =>,

• A set u of propositions will be called explanatory closed, if u C C(u). • A theory u of => will be called exact, if it is consistent and explanatory closed, that is, u = C(u). • A set u of propositions is an exact theory of a causal theory A, if it is an exact theory of =Â • An exact theory describes an informational state in which every proposition is produced, or explained, by other propositions accepted in this state. Accordingly, restricting our universe of discourse to exact theories amounts to imposing a kind of an explanatory closure assumption on a production relation. Namely, it amounts to requiring that any accepted proposition should also have reason, or explanation, for its acceptance. This suggests the following notion of a nonmonotonic semantics: Definition 8.6 A nonmonotonic production semantics of a production inference relation or a causal theory is the set of all its exact theories.


243

The nonmonotonic production semantics, as defined above, is a certain set of propositional theories. As for biconsequence relations, this abstract definition leaves us much freedom in determining what can be seen as nonmonotonic consequences of a causal theory. The most obvious choice consists in taking propositions that belong to all exact theories. As we will see, however, this means that we accept only propositions that belong to the least exact theory. But we can also choose propositions that belongs to all maximal exact theories. Similarly, we can consider all propositions that do not belong to any such theory as nonmonotonically rejected. All these options determine useful variants of a sceptical, or cautious, nonmonotonic semantics. A credulous nonmonotonic semantics can be defined by considering propositions that belong (or not belong) to at least one exact theory. As studies in nonmonotonic reasoning show, each of these options could be appropriate for particular reasoning tasks. Again, it could be safely argued that all the information that can be discerned from the nonmonotonic production semantics of a causal theory, or a production inference relation, can be seen as nonmonotonically implied by the latter. The nonmonotonic production semantics for causal theories is indeed nonmonotonic in the sense that adding new rules to the production relation may lead to a nonmonotonic change of the associated semantics, and thereby to a nonmonotonic change in the derived information (a number of examples will be presented below). This happens even though production rules themselves are monotonic, since they satisfy Strengthening (the Antecedent). Exact theories are precisely fixed points of the production operator C. Since the latter operator is monotonic and continuous, exact theories (and hence the nonmonotonic semantics) always exist. Also, the general properties of monotonic operators imply the following properties of exact theories: Lemma 8.16 (1) Any production relation has a least exact theory. (2) Any theory of a production relation contains a greatest exact theory. (3) The union of any chain of exact theories (with respect to set inclusion) is an exact theory. (4) Any exact theory is included in a maximal exact theory. Proof. (1) The least fixed point of C determines a least exact theory. (2) If C(u) C w, then iterating C starting from u will give us a greatest fixed point of C that is included in u.

244


(3) Let {ui} be a set of exact theories of => linearly ordered by inclusion, and u = {Ju{. Since each M,- is a theory of =>, the usual compactness argument implies that u will also be a theory of =>, that is C{u) C u. In addition, by the monotonicity of C, we immediately have that u is explanatory closed, that is, u C C(u). Hence u = C(u). (4) Follows from (3). • Unfortunately, the following example shows that exact theories are not closed with respect to arbitrary intersections. Example 8.1 Let us consider a causal theory A = {p; => p;, p,-=$> g | i > 0}. Then, for any natural n, the set un — Th.(q,pn) is an exact theory of =>A- However, (~) un = Th(^) is not an exact theory of =>AAs a result, a least exact theory containing a given set of propositions does not always exist. The following simple lemma gives a constructive description of the nonmonotonic production semantics of a causal theory. The proof is immediate by Proposition 8.6. Lemma 8.17

u is an exact theory of a causal theory A iffu = Th(A(u)).

We are going to show now that regular production inference provides an adequate and maximal logical framework for reasoning with exact theories. Definition 8.7 Two causal theories will be called (strongly) nonmonotonically equivalent if they are (strongly) equivalent with respect to the nonmonotonic production semantics. Note that production inference relations can also be considered as (rather big) causal theories. As before, let =$-rA denote the least regular production relation that contains a causal theory A. Then we have Lemma 8.18

A causal theory A is nonmonotonically equivalent to =>^.

Proof. If v is a theory of A, then v — CnA(v), and hence Th(A(v)) = Th(A(CnA(u))). Consequently, v = Th(A(u)) iff v = Th(A(CnA(«))). By Corollary 8.15, this implies that v is an exact theory of A if and only if it • is an exact theory of =>•A . The above lemma implies that the postulates of regular inference are adequate for reasoning with exact theories, since they preserve the latter. Moreover, we will show now that regular inference relations constitute a maximal logic suitable for the nonmonotonic production semantics.


245

Let us say that causal theories A and F are regularly equivalent, if each can be obtained from the other using the postulates of regular production inference. Or, equivalently, when =>A = =>£.. Strong nonmonotonic equivalence can be seen as an equivalence with respect to the background monotonic logic of causal theories. And the next result shows that this logic is precisely the logic of regular production relations. Theorem 8.19 Two causal theories are strongly nonmonotonically equivalent if and only if they are regularly equivalent. Proof. To simplify notation, =>$ below will denote the least regular production relation containing a causal theory \t, while C

Explanatory Nonmonotonic Reasoning

Explanatory Nonmonotonic Reasoning

Readings in Nonmonotonic Reasoning

Readings in Nonmonotonic Reasoning

Logic Programming, Knowledge Representation, and Nonmonotonic Reasoning

Logic Programming and Nonmonotonic Reasoning - LPNMR 2011

Logic Programming and Nonmonotonic Reasoning, 9 conf., LPNMR 2007

Handbook of logic in AI and logic programming, vol.3: nonmonotonic reasoning and uncertain reasoning

God as Self - Explanatory

Conditionals in Nonmonotonic Reasoning and Belief Revision - Considering Conditionals as Agents

IELTS practice tests: with explanatory key

For the Sake of the Argument: Ramsey Test Conditionals, Inductive Inference and Nonmonotonic Reasoning

Critical Reasoning

Reasoning Practically

Seductive Reasoning

Commonsense Reasoning

Practical reasoning

Seductive Reasoning

Critical Reasoning

Hypothetical Reasoning

Nonmonotonic Logics: Basic Concepts, Results, and Techniques

An Absurd Reasoning

Similarity and Analogical Reasoning

Elements of Logical Reasoning

Reasoning About Uncertainty

Neural-symbolic cognitive reasoning

Reasoning about Uncertainty

Introduction to Reasoning

Reasoning Skills Success

Jewish Liturgical Reasoning

Similarity and Analogical Reasoning

Explanatory Nonmonotonic Reasoning